Boris Johnson said on Thursday that he would step down as Britain’s prime minister, after a wholesale rebellion of his cabinet, a wave of government resignations and a devastating loss of party support prompted by his handling of the party’s latest sex-and-bullying scandal.

Mr. Johnson said he planned to stay on in his post until the Conservative Party chooses a new leader, which could take several months. He said he expected the timetable for his departure, and the selection of a successor, to be set on Monday by a committee of senior Conservative lawmakers.

“It is clearly now the will of the parliamentary Conservative Party that there should be a new leader,” Mr. Johnson said in remarks outside Downing Street. “The process of choosing that new leader should begin now.”