The Black Sabbath singer said he was "mortified" at the effect his behaviour had had on his family.
Osbourne, 67, made the statement following an interview given by his hair stylist Michelle Pugh.
Ms Pugh told People magazine she had had a "very real relationship" with the rock star which had lasted four years.
In his statement, the rock singer said: "Over the last six years, I have been dealing with a sex addiction.
"I'm sorry if Ms Pugh took our sexual relationship out of context.
"I'd also like to apologise to the other women I have been having sexual relationships with."
He added: "Out of bad comes good. Since the press exposed this, I have gone into intense therapy."
The singer and his X Factor judge wife have been married since 1982.
Their often chaotic family life with children Jack and Kelly was the focus of a hit fly-on-the-wall MTV series from 2002 to 2005.
But earlier this year, the singer moved out of the family home.
His wife subsequently told US TV chat show The Talk  she had "no idea" if their marriage had a future.
However, the couple later said they were back together.
In his statement, Osbourne paid tribute to his wife for standing by him.
"I thank God that my incredible wife is at my side to support me," he said.
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Ozzy Osbourne has revealed he is undergoing "intense therapy" for a "sex addiction" that was putting his marriage to Sharon Osbourne at risk.