LONDON — By the arcane protocols governing monarchs and ministers, it was a moment that should never have happened.

Caught by a television crew’s microphone, Prime Minister David Cameron of Britain was overheard during a visit to New York this week telling the former mayor, Michael R. Bloomberg, that Queen Elizabeth II “purred” when he called to inform her that Scottish voters had rejected independence in a referendum.

“The definition of relief, if you are prime minister of the United Kingdom, is ringing up Her Majesty the Queen and saying ‘Your Majesty, it is all right, it’s O.K.’ ” Mr. Cameron told Mr. Bloomberg. “That was something. She purred down the line.”

The language aside — purring on the phone is usually associated with a different kind of communication — Mr. Cameron’s remarks were a kind of undiplomatic double whammy.