Gilgamesh dives into the sea with stones attached to his feet and brings up the ''Flower of Immortality.'' In a clear parallel with the biblical story of Adam and Eve, he allows a serpent to eat the flower, cheating mankind of its benefits.

Other ancient records show that 4,000 years ago Dilmun was also a great trading center and the capital of an empire. Yet until 1857 when Sir Henry Rawlinson, a British scholar, discovered how to read cuneiform and the first references to this ancient island paradise were deciphered, the name of Dilmun had vanished from the collective memory of mankind.

That Nineveh, Babylon, Thebes and Ur had all once been great cities was never entirely forgotten because their names are recorded in the Bible, even if little was known about them. But for thousands of years the legend of Holy Dilmun, the island paradise where man lived forever, disappeared.

The enthralling tale of how Geoffrey Bibby and other archeologists from the Prehistoric Museum in Aarhus, Denmark, identified the present-day Persian Gulf island of Bahrain as the site of the lost paradise of Dilmun has been told by Mr. Bibby in his book ''Looking for Dilmun.'' It is one of the most gripping archeological detective stories ever written.

But visitors to Bahrain today can still relive Mr. Bibby's search for Dilmun with the help of his book, visiting the major excavation sites, inspecting the most important finds and recapturing something of the excitement he felt as the pieces in the puzzle slowly fell into place and the ruins of ''lost'' Dilmun emerged from beneath his trowel.

Today this tiny, verdant island, with its copious freshwater supplies, still seems a kind of paradise in that parched, bone-dry region of the world. Lying just off the coast of Saudi Arabia, barely 60 miles long and 30 wide, Bahrain is the garden of the Persian Gulf, with shady palm groves full of bright- colored birds, fruit and flowers.

At several points around the coast, springs bubble up through the brine from the ocean floor with such force that you can drop a bucket into the sea and pull it up full of fresh water.

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It remains a famous center for pearl fishing, where divers still weight their feet

with stones just as Gilgamesh did. And

how much more beautiful than modern

white cultured pearls are the big, lumpy natural ones, with their luminous, greenish hue.

Emerald shoal waters surround the island. And at the end of the last century, Captain (later Sir) Edward L. Durand, an early explorer, described daybreak on Bahrain in terms still true today. Nature, he wrote, seemed to have ''exhausted every tint of living green in her paint box; and then, wearying of the effect, splashed a streak of angry purple into the foreground.''

But Bahrain has a mysterious side. In the sandy center of the island lies a truly astonishing sight - thousands upon thousands of little hillocks, each some 10 to 15 feet high and packed tightly together in an unending vista so that at first the visitor thinks they must be a natural phenomenon, like sand dunes. But they are not. The hillocks are manmade grave mounds. Bahrain, the original Garden of Eden, is also an ancient island necropolis.

In another sense, too, Bahrain remains a Persian Gulf paradise. For it is the only Gulf state that welcomes Western tourists and tries to make them feel at home. The island has been designated by its neighbors as the region's financial and services capital, the place where banks, accountancy firms and engineering companies working in the area are encouraged to establish their headquarters and base Western employees.

Although a Moslem country like other Gulf states, Bahrain is more tolerant than Saudi Arabia or Kuwait. Alcoholic drinks, pork dishes, magazines and books shunned elsewhere in the Gulf are all available in hotels and restaurants frequented by foreigners. Women are permitted to drive, and mixed swimming is allowed.

These days the hunt for Dilmun begins at Bahrain's National Museum, over the causeway from the capital of Manama on Al- Muharraq island near the airport.

This is an essential first stop because visitors need to get a pass from the director, Sheika Haya Al-Khalifa, to visit the Barbar Temple, a key Dilmun relic. But the museum also contains a splendid display of Geoffrey Bibby's major finds, telling the story of the discovery of Dilmun and its links with the Garden of Eden.

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Here, too, is a replica (the original was lost in London during World War II) of the first major clue identifying Dilmun with Bahrain. It is a foot-shaped black basalt stone found by Captain Durand in 1879 while surveying the island's antiquities and bearing the Sumerian inscription ''Palace of Rimum, slave of the God Inzak, Man of the tribe of Agarum.''

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In a masterly paper, Sir Henry Rawlinson, then 75 years old, showed that Inzak, or Enzak, was the titular god of Dilmun. After surveying all known references to Dilmun in ancient Sumerian literature, he argued for the first time that Bahrain must be the site of the island Paradise, last described by King Sargon of Assyria before it faded from human memory as ''a fish in the midst of the Sea of the Rising Sun.''

But if Sir Henry, working from written evidence, was right and Bahrain was Dilmun, then, Mr. Bibby reasoned, remnants of its lost civilization must still be buried there. He had only to dig.

Besides the Dilmun antiquities, the museum has displays devoted to pearl fishing and local art and clothes. Particularly interesting are those devoted to women's clothing, showing the sumptuous traditional Bahraini marriage dress and the scarcely less magnificent clothes worn in the old days by unmarried women. Of course, in the street, Bahraini women to this day continue to be shrouded in black veils, but at home they look very different.

A 15- minute taxi ride from any of the big hotels along King Faisal Boulevard will take the Dilmun enthusiast to the site of Mr. Bibby's first great find: Qala'at al-Bahrain, known in English as the Portugese fort.

Only the tumbled-down walls remain, though inside you can see the remnants of the palm-leaf encampment Mr. Bibby and his Danish companions built there in 1957. the last vestiges of what the acheologists dubbed the ''Carlsberg Culture'' after the Danish beer they drank.

But below the fort's southern walls lie remaining bits of the walls of the Dilmun-age ''palace'' they found, the first evidence there was a thriving civilization on the island 4,000 years ago. Here were discovered the famous Dilmun seals and weights, which link the island with ancient Ur and the Indus Valley, confirming its importance as a trading center. But here Mr. Bibby also found astonishing evidence of a direct religious link between the inhabitants of ancient Dilmun in the third millennium B.C. and the much older legend of Gilgamesh's unfortunate encounter with the serpent - ritually buried rows of pots, each containing a snake's skeleton and a pearl.

The pearl Mr. Bibby identifies with the Flower of Immortality, recalling how in ancient Egypt Cleopatra drank an elixir of pearls dissolved in wine. ''Here we have clear proof that the legend of Gilgamesh was still a living and integral part of the religion of Bahrain at the time the palace was built and inhabited,'' he writes.

Today, visitors can scramble into the pit and walk along the streets of ancient Dilmun, admire the great stone doorway of the palace, clamber through its rooms and recall how its inhabitants also commemorated man's fall centuries before Genesis was written.

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A few miles west of the fort lies Mr. Bibby's second major site, the Barbar Temple. After their first exploration, Mr. Bibby and his companions re-covered the temple with sand to stop theft. But last year the Bahrain Government re-excavated it, building a permanent site with concrete walkways and viewing points for visitors.

The first of three temples was built here nearly 5,000 years ago. But as the temple complex grew bigger over the centuries, a magnificent sacred well was always retained as its central feature. Well-trod steps lead down to a limpid pool of fresh, blue water contained in finely cut limestone blocks. Nearby stands a stone altar with a drain for the blood of sacrificial animals.

For most scholars the Barbar Temple was sacred to Enki, Lord of the Abyss and Ruler of the Sweet Waters Under the Earth, the god who saved mankind from the Great Flood and who lived in Holy Dilmun at the beginning of time. Its well was thus a sacred link with the god holding dominion over the fresh waters on which life depended.

And sure enough, an ancient Sumerian fragment refers to a great Temple of Enki, calling it ''the far-famed house'' that is ''built in the heart of the Lower Sea,'' the name then given to the Persian Gulf. Bahrain is half- way down the Gulf, in ''the heart of the Lower Sea.''

Close by, near Diraz village, lies an intriguing mystery Mr. Bibby never resolved. The well here, once the biggest on Bahrain, was filled in around 800 A.D. by a Moslem ruler to punish the villages for idolatry. Today, stone blocks used to seal the well litter the sand. Visitors can also see the ''tantalizing stairway'' Mr. Bibby found leading down into the well, and where he discovered two broken idols from Dilmun times. Was there another sacred well here where the villagers still worshiped Enki 3,000 years after the Barbar Temple was built and 800 years into the Christian era? We don't know. The excavation was never completed.

Only two stops remain in the quest for Dilmun. First to Ali village to see the ''Royal Mounds,'' a cluster of unusually large grave mounds where Mr. Bibby found the cups and glasses he called ''a table service worthy of the Kings of Dilmun.''

Anywhere along the road, a visitor can stop and inspect the island's grave mounds, many now opened by archeologists. They housed the dead citizens of Dilmun, and the discovery of similar burial mounds in Saudi Arabia, Qatar and Failaka Island off Kuwait helped Mr. Bibby establish the boundaries of Dilmun's empire.

After this long, hot tour, visitors may want to drive another few miles to the Emir's beach at Az Zallaq on the west coast. Here Bahrain's ruler, Sheik Isa bin Sulman Al Khalifa, keeps a marble summer palace on a silver beach edged with shady lawns and golden lampposts. He allows Western visitors to swim there free of charge, eat picnics under the palm trees and stroll on a black and gold pier above an emerald sea. It is a present-day paradise, a modern Dilmun.

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Tips for touring Bahrain

The best time to visit Bahrain is from January to June, before the humid summer heat sets in. United States citizens are granted 72-hour visas upon arrival in Bahrain; permits for longer stays are given only to persons doing business in the country.

Bahrain is full of luxurious air-conditioned hotels, complete with swimming pools. One recommendation is the Inter-Continental, also known as the Regency. It is within easy walking distance of the souk, or Arab market, and of the business center of Manama. The others tend to be a taxi drive away. None of them is inexpensive: A double room is around $120 a night. An average meal easily runs $60 a person.

The best way of traveling around the island is in a taxi or a chauffeur- driven rented car, making sure you get a knowledgeable, English-speaking driver. A half-day with a rented car and chauffeur costs about $90. Taxis have no meters, so you haggle over the price. A trip to the airport is around $20 to $30; a half-day's driving around, about $60 to $75.

The National Museum is open from 9 A.M. to 2 P.M. every day but Friday. Admission is free. The telephone number is 320283. P. L.