“There are no truth to the allegations,” he said. “If you go back 20 years ago, they were investigated. Law enforcement obviously didn’t feel there was merit there. For me, I’ve just tried to move on and go forward.”

The Phoenix Police Department investigated the claim but closed the case without filing charges. Mr. Johnson entered a settlement that included a $230,000 payment to his accuser. The mayor declined to comment when asked why, if the charges were unfounded, he had agreed to make a payout.

What is different this time is a videotape of a police interview in which the girl, speaking in a calm voice, describes in disconcerting detail what she said Mr. Johnson did to her, including groping and undressing her. The release of the video, posted on Deadspin, a website devoted to sports news, put a young and vulnerable voice to claims that until now had been only a detached account found in court filings.

“People have so desperately wanted to believe in him that they’ve given him a pass on a lot of things, and I think that has worked out generally to the city’s advantage,” Steve Hansen, a Democratic City Council member and critic of Mr. Johnson’s, said before the announcement. “The wall of not knowing how to deal with his problems, but choosing to ignore the failures in hopes of the promise, has begun to collapse. Once you saw the video and read the story, it was hard to pretend it never happened.”

Mr. Johnson, 49, has long been a polarizing force in this city.

For his admirers, he is the vibrant face of a new Sacramento — epitomized by the $500 million arena going up on J Street, surrounded by a burst of construction inspired by the project. He is a former president of the United States Conference of Mayors and a regular guest of President Obama at the White House. He is married to Michelle Rhee, the high-profile school superintendent who, like Mr. Johnson, has a history of battling teachers’ unions.