Edwards, 55, said he told his entire family about the affair after it ended in 2006, and that his wife Elizabeth, who has incurable breast cancer, was "furious" but that their marriage would survive. The couple have three children, Cate, 25, Emma, 9, and Jack, 7. When he confessed his affair to his wife, "she was mad," Edwards said.
"She was angry," he said. "I think 'furious' would be a good way to describe it. She didn't understand. We both went through a process of trying to figure out how it happened, why it happened." Late Friday evening, Elizabeth Edwards posted a message on her blog, saying that she and her children will stand by Edwards.
"Our family has been through a lot. Some caused by nature, some caused by human weakness, and some - most recently - caused by the desire for sensationalism and profit without any regard for the human consequences," she wrote on http://elizabethedwards.dailykos.com/. "None of these has been easy. But we have stood with one another through them all. Although John believes he should stand alone and take the consequences of his action now, when the door closes behind him, he has his family waiting for him."
She too, denied in the blog post that Hunter's child was fathered by her husband.
Edwards told ABC News that he met secretly with former lover Rielle Hunter as recently as last month in a California hotel room at her request because "she was having some trouble, she just wanted to talk."
"I wanted her not to tell the public what had happened," he said at another point. "Very simple. That's the reason I went."
Edwards contacted ABC News and requested that he be interviewed about the allegations. He agreed before the interview to place no limitations on the questions he would be asked, but limited the amount of time he would be questioned. He declined to clarify exactly when the relationship began or ended but said it was over before he announced his campaign for the presidency on Dec. 28, 2006.
"I think my family is entitled to every detail," Edwards said. "They've been told every detail…I think beyond the basics…I think that's where it stops in terms of the public because I think everything else is within my family and those privacy boundaries out to be respected."
A series of dramatic allegations revolving around the affair have been reported by the supermarket tabloid National Enquirer. The most serious of those allegations is that Hunter has been receiving hush money through the Edwards campaign in order to keep her quiet.
In the interview, Edwards denied knowing anything about any support being provided to Hunter or to Andrew Young, the former Edwards campaign aide who has said that he, not Edwards, is the child's father.
If any such support was being provided, Edwards insisted, "I had nothing to do with any money being paid, and no knowledge of any money being paid, and if something was paid, it wasn't being paid on my behalf."
Edwards' 2008 national finance chairman Fred Baron said late Friday that he had provided "assistance" to Hunter and others without Edwards' knowledge.
In a statement emailed to ABC News, Baron said that he "decided independently to help two friends and former colleagues rebuild their lives when harassment by supermarket tabloids made it impossible for them to move forward on their own." "I did this of my own volition without the instruction or suggestion of anyone, and made a conscious decision not to tell anyone, including John Edwards, that assistance was provided. The assistance was offered and accepted without condition. This is now and shall always remain a private matter between these individuals and me."
Edwards said he knows he could not be the father based on the timing of the baby's birth on February 27, 2008.
A grainy photo that appears to show Edwards holding a child was published recently in the National Enquirer, which claimed that the baby was Rielle Hunter's child Frances Quinn. Edwards said he couldn't make "sense" of the photo.
"I don't know if that picture is me," Edwards said. "It could well be. It looks like me. I don't know who that baby is. I have no idea what that picture is."
Pressed to clarify whether the child in the picture was Hunter's child, Edwards suggested that the tabloid may have digitally altered the picture, but also noted the frequency with which politicians hold babies. "Do you know how many pictures have been taken of me holding children in the last three years? I mean it happens all the time."
Edwards was asked again about the meeting.
"Are you saying you don't remember holding that child of Miss Hunter?"
"I'm saying you asked me about that photograph. I don't know anything about that photograph, I don't know who that baby is. I don't know if the picture has been altered, manufactured, if it's a picture of me taken some other time, holding another baby…I have no idea. I was not at this meeting holding a child for my photograph to be taken I can tell you that."
Edwards finally told Woodruff that he did not remember a baby being present at the meeting last month in California.
Edwards blamed the affair on the adulation surrounding his remarkable rise into presidential politics.
"I went from being a senator, a young senator to being considered for vice president, running for president, being a vice presidential candidate and becoming a national public figure. All of which fed a self-focus, an egotism, a narcissism that leads you to believe that you can do whatever you want. You're invincible. And there will be no consequences."
According to friends of Hunter, Edwards met her at a New York city bar in 2006. His political action committee later paid her $114,000 to produce campaign website documentaries despite her lack of experience. Edwards said the affair began during the campaign after she was hired. Hunter traveled with Edwards around the country and to Africa.
When the National Enquirer first reported the alleged Edwards-Hunter affair last October 11, Edwards, his campaign staff and Hunter vociferously denounced the report.
"The story is false, it's completely untrue, it's ridiculous," Edwards told reporters then.
He repeated his denials just two weeks ago.
Edwards today admitted the National Enquirer was correct when it reported he had visited Hunter at the Beverly Hills Hilton last month.
Edwards made a point of telling Woodruff that his wife's cancer was in remission when he began the affair with Hunter. Elizabeth Edwards has since been diagnosed with an incurable form of the disease.
Edwards was asked whether he loved Hunter.
"I'm in love with one woman," he responded. "I've been in love with one woman for 31 years. She is the finest human being I have ever known."
The former senator was asked why his wife was not by his side as he made the admissions about his affair to ABC News.
"I have seen these public figures who bring their wives along when they say they've done something wrong; confessing some wrong and the wife – they bring their wives to stand beside them I guess, to show support…Elizabeth didn't do anything wrong. I…didn't ask for her to be here. I asked for her not to be here. Uh, this is my responsibility; my alone. I have to be the man and take responsibility."
Earlier in the interview, Edwards described his experiences in personal, visceral terms.
"I am imperfect," he said. "And anybody, anybody watching this broadcast or who hears about this who wants to beat me up for this, they should have at it. The truth is – you can't possibly beat me up more than I have already beaten myself up."