LONDON — His jacket neatly buttoned, his hair corralled into chaos-free submission, Boris Johnson stood outside Downing Street on Wednesday and in his first moments as prime minister promised to do what so far has proved impossible: lead Britain through an orderly, on-time exit from the European Union.

“The doubters, the doomsters, the gloomsters — they are going to get it wrong,” he declared. “The people who bet against Britain are going to lose their shirts.”

It was a supremely confident, at times grandiloquent address from a man whose silver-tongued talk often surpasses his performance. After a lifetime of joking and blustering and maneuvering his way into jobs and then sabotaging himself with poor preparation and deceitful behavior, Mr. Johnson, 55, seems determined to prove he can put aside his court-jesterish ways and rise to the occasion.

He seemed unbothered by the fact that he is entering office with a paper-thin working parliamentary majority at a time of deep divisions within the country and in his own Conservative Party.