In a statement issued this afternoon, Dr O'Mahony says he has been considering his position for some months.
He said that in light of the unfavourable criticisms of him in the Murphy report and the continuing public controversy about them he has decided to resign from the charity with immediate effect.
Bishop O'Mahony said he had delighted memories of his work for the trust which had sustained him for a lifetime and he prayed for its success in the future.
The Murphy Commission found that Bishop O'Mahony's response to allegations of child sexual abuse was worse than that of any other living auxiliary bishop of Dublin.
A spokesperson for Bishop O'Mahony said he will not be making any public comment due to ill health.
The former Auxiliary Bishop of Dublin resigned in 1996 due to a heart condition.
Meanwhile, a statement from the Vatican press office said the planned talks with Cardinal Seán Brady and Archbishop Diarmuid Martin will inform the Pope about what it described as the 'sorrowful situation' of the church in Ireland.
The head of the Congregation for the Doctrine of the Faith, Cardinal Joseph Levada, is expected to be among at least six Prefects of Vatican congregations who will accompany Pope Benedict XVI in his planned meeting with Cardinal Brady and Archbishop Martin.
The meeting takes place following the publication of the Murphy Commission report on clerical child sex abuse in the Dublin Catholic Archdiocese.
The Papal Nuncio Archbishop Giuseppe Leanza will also attend, according to the statement.
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SIC: RTÉ