Mr Ford told reporters at city hall he had taken the drug about a year ago but was not an addict.
The mayor has rejected calls for his resignation after the revelation that a video apparently showing him smoking crack was in police possession.
He has defied calls to quit, vowing to run for re-election next year.
"Yes, I have smoked crack cocaine," Mr Ford said outside his office on Tuesday.
"There have been times when I've been in a drunken stupor. That's why I want to see the tape.
"I want everyone in the city to see this tape. I don't even recall there being a tape or video. I want to see the state that I was in."
'It is what it is'
But Mr Ford maintained to reporters that when he previously denied using the drug: "I wasn't lying. You didn't ask the correct questions."
Six months ago, the mayor said: "I do not use crack cocaine, nor am I an addict."
On Tuesday, he said: "I made mistakes in the past and all I can do is apologise. But it is what it is and I can't change the fact, and I can apologise to my family, my friends, my colleagues and the people of this great city."