Iris Robinson also admitted she encouraged friends to provide financial backing to assist the man in a business venture.
The relationship 18 months ago was a brief one but in a dramatic statement she revealed the devastating impact it had on the family and how she attempted to take her own life last March.
She said: "Everyone is paying a heavy price for my actions ... I am so, so sorry."
The man has not been identified, and even though Mr Robinson considered walking away because of his sense of betrayal, the First Minster insisted tonight he was determined to save his marriage, would be back at his desk in Belfast tomorrow, and was staying on as leader of the Democratic Unionist Party going into this year's general election.
Sitting in the study of his home on the outskirts of east Belfast, choking with emotion and with his eyes welling up, Mr Robinson said: "That is the road I am on. It is a road without guarantees, but not without hope."
Nine days ago Mrs Robinson, MP for Strangford and a member of the Northern Ireland Assembly, declared she was quitting politics because of a battle with depression. But since then there has been intense behind-the-scenes speculation about the state of their 40-year marriage.
Mr Robinson has been away from his office at Parliament Buildings, Stormont, but suddenly called a special press briefing this afternoon at his house in Dundonald where officials issued a personal statement by his wife - before he went on to speak to four journalists to confirm the couple's private turmoil.
He said: "I love my wife. I have always been faithful to her. In a spirit of humility and repentance, Iris sought my forgiveness. She took responsibility upon herself alone for her actions and I have forgiven her. More important, I know that she has sought and received God's forgiveness."
Mother-of-three Mrs Robinson's admission of an affair and her husband's extraordinary appearance in front of the TV cameras will stun Northern Ireland. Even his closest colleagues and political rivals will be left shocked at such a public confession.
The marriage was considered by all sides to be rock-solid, with Mr Robinson frequently rallying to his wife's defence when her outspoken criticism of homosexuality saw many label her a political liability.
Mrs Robinson was not to be seen at her luxury home today, but in her penitential statement she spoke about how severe bouts of depression had altered her mood and personality.
She said: "I fought with those I loved most, my children and friends; saw plots where none existed and conducted myself in a manner which was self-destructive and out of character.
"During this period of mental illness I lost control of my life and did the worst thing that I have ever done. Over a year and a half ago, I was involved in a relationship.
"It began completely innocently when I gave support to someone following a family death. I encouraged friends to assist him by providing financial support for a business venture.
"Regrettably, the relationship later developed into a brief affair. It had no emotional or lasting meaning, but my actions have devastated my life, and the lives of those around me."
Mrs Robinson, 60 and an MP since 2001, added: "Everyone is paying a heavy price for my actions. Psychiatrists may suggest that my mental illness was a significant factor explaining my irrational behaviour.
"I do not in any way question or doubt their judgement. But in order to master my life, I do not want to dilute the blame or resist taking full responsibility for my actions. I am completely ashamed and deeply embarrassed."
She said she had hurt her husband, family and friends, let down thousands of people who had placed their faith in her and - though her medical condition was a factor - she had not been true to the values she professed.
"I would pay any price on Earth to take back the wrong I have done and the hurt I have caused to those around me," she said. "I love my husband more than I can ever say. I know this more now than ever before."
And it was at this stage that she disclosed how at midnight on March 1 last year - believed to be at their home - she tried to take her own life when she confessed to the affair because of the guilt she felt.
Just hours later her husband was at the dispatch box answering questions at the Northern Ireland Assembly.
Mrs Robinson said she was determined to regain her health and work to repair the damage to her marriage. She added: "I do not deserve a second chance, but I have been given one. Nothing is more important to me.
"I sincerely apologise to all those I have hurt and let down. I have inflicted deep pain on my husband Peter, my family, friends, staff and all those who have supported me. I am so, so sorry."
Mr Robinson claimed the couple first considered going public about the affair last November, but decided against the move because of medical advice that his wife would not have been able to cope with the consequences.
He felt devastated and deeply hurt in the most difficult period of his life. He felt the pain of it every day.
He said: "My immediate impulse was to walk away from my marriage. I felt betrayed after almost 40 years of being happily and closely bonded together. The circumstances I faced, however, caused me to take a different course.
"Iris, racked with guilt and sorrow, had attempted to take her own life and would certainly have been less likely to recover if I had left. Over time and on calmer reflection, I set her inappropriate behaviour against 40 years of bringing up our children - often alone.
"Forty years of selflessly giving me the space to pursue my beliefs; of facing public pressure for the stand I was taking and having to live with the threats and dangers my position visited upon my family.
"Those were 40 years during which she supported me more than any person could reasonably have been expected to. Forty years where we walked the valley basin as well as the mountain paths. But most of all, 40 years during which we shared a strong loving relationship."
He said he was determined to save the marriage, re-build their lives and put the issue behind him.
Mr Robinson added: "That is the road I am on. It is a road without guarantees, but not without hope. I love my wife. I have always been faithful to her.
"In a spirit of humility and repentance Iris sought my forgiveness. She took responsibility upon herself alone for her actions and I have forgiven her. More important, I know that she has sought and received God's forgiveness."
If people must judge, then he said they should do so with mercy and compassion.
"It is heartbreaking that all the good work she has done for so many over decades will be overshadowed by what she has outlined today."
Mr Robinson, who is due to meet Deputy First Minister Martin McGuinness tomorrow to discuss the deadlock over the delay in transferring policing and justice powers from London to Belfast - a hold-up which is threatening the future of the power-sharing executive - also spoke of speculation about his financial affairs.
He told the Press Association: "In all my public life I have acted in the most professional and ethical way. There is nothing that anybody will produce.
"I have heard all sorts of nonsense and rumours over the last number of days, even indicating that I had lost £3 million in shares and a lot of other stuff as well. Absolute rubbish."
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