Richard Burke stepped down from the Archdiocese of Benin in Nigeria earlier this year after a woman, aged in her 40s and now living in Canada, made a complaint against him.
The St Patrick's Missionary Society in Kiltegan, Co Wicklow, which the Archbishop was a member of before being appointed to the hierarchy, confirmed his resignation had been accepted by the Vatican.
No findings have been made against the Archbishop, one of the most senior Catholic clerics to face allegations of child sex abuse.
In a statement the Kiltegan fathers said Archbishop Burke had admitted breaking his vow of celibacy.
"He has apologised to all those whom he has hurt by his actions and has taken full personal responsibility for what he has done wrong," the society said.
A complaint was made in December 2008 by Dolores Atwood alleging she had been sexually abused when she was 14-years-old by Archbishop Burke, then a priest ministering in Warri Diocese, Nigeria.
She claimed she had an ongoing relationship with him afterwards.
Archbishop Burke denied the allegation of child sex abuse but admitted having the affair when Ms Attwood was over 18.
The Congregation for the Doctrine of the Faith in Rome investigated the allegations against the Archbishop.
"The investigation of the allegation carried out by our Child Protection Officer in Nigeria found no evidence to corroborate the allegation of child sexual abuse made by Mrs Atwood," the society said.
"The relevant civil and ecclesiastical authorities in Nigeria were informed of the allegation. We have had no further communication from the Congregation for the Doctrine of the Faith."
The society said it was still in touch with Ms Attwood through a third party.
"We are aware that she and her family continue to suffer from the consequences of the relationship," it said.
"We have recently agreed to renew our offer of counselling and have expressed again our deep sorrow for the suffering she and her family are going through."
In a statement to the Irish Catholic newspaper, Archbishop Burke said he was deeply ashamed.
"The reason for my resignation is that I have been unfaithful to my oath of celibacy. I am deeply ashamed of my behaviour. I have caused great pain, hurt and bewilderment to those around me," he said.
"I take full responsibility for my actions. I wish to express my deepest sorrow for my inappropriate, irresponsible and repeatedly sinful conduct."
The Archbishop said he was told in January 2009 about the abuse allegation.
"I responded immediately that I have never, ever, in my life - in any way - sexually abused a child. This is still my position. It is the truth," he said.
The society told him there was no evidence to back up the abuse allegations.
"I apologise sincerely for the pain and hurt I have caused Mrs Dolores Atwood and her family," the Archbishop said.
The cleric said he and Mrs Atwood began a relationship in the latter part of 1989, when she was 21 and he was 40.
The Archbishop returned to Ireland in March the following year and it was six years before he went back to Nigeria.
"In the last 20 years, Mrs Atwood and I met on seven occasions. On three of those occasions our relationship was again expressed sexually. This was entirely inappropriate behaviour and it is something for which I am truly sorry," Archbishop Burke said.
"I apologise to my family, to my relatives, to my friends and to my fellow members of St Patrick's Missionary Society.
"I also ask the forgiveness of the people, religious and clergy of the Diocese of Warri and the Archdiocese of Benin City. I am very aware that I have hurt many people and I ask their forgiveness and prayers."