“Our country needs the left, but a modern, innovative left,” he said.

Mr. Hamon’s victory was the clearest sign yet that voters on the left want a break with the policies of President François Hollande, who in December announced that he would not seek re-election. However, Mr. Hamon’s strong showing is unlikely to change widespread assessments that left-leaning candidates have little chance of making it into the second round of voting in the general election.

The first round of the general election is set for April 23 and the runoff for May 7.

The Socialist Party is deeply divided, and one measure of its lack of popular enthusiasm was the relatively low number of people voting. About two million people voted in the second round of the primary on Sunday, in contrast with about 2.9 million in the second round of the last presidential primary on the left, in 2011.

However, much of the conventional wisdom over how the elections will go has been thrown into question over the past week, because the leading candidate, François Fillon, who represents the main right-wing party, the Republicans, was accused of paying his wife large sums of money to work as his parliamentary aide. While nepotism is legal in the French political system, it is not clear that she actually did any work. Prosecutors who specialize in financial malfeasance are reviewing the case.

France’s electoral system allows multiple candidates to run for president in the first round of voting, but only the top two vote-getters go on to a second round.