Archbishop Diarmuid Martin has sent a letter to all priests in the Dublin Diocese to confirm the news.
The two auxiliary bishops submitted their resignations to Pope Benedict on Christmas Eve 2009, in the wake of the Murphy Report investigation into clerical child abuse in the Dublin Diocese.
The two are to be assigned revised responsibilities within the diocese.
Dr Walsh has been a bishop for 20 years and ran the diocese of Ferns for four years after Brendan Comiskey resigned as a bishop over his cover-ups of abuse.
Dr Field has served for 12 years in Dublin.
Both bishops initially resisted calls for their resignation.SIC: RTÉ
The resignations of two Dublin auxiliary bishops, announced in the wake of the Murphy report into clerical child abuse, have been rejected by Pope Benedict XVI.
Bishop Raymond Field and Bishop Eamonn Walsh tendered their resignations on Christmas Eve 2009 after coming under intense pressure because they had served as bishops during the period investigated by the Murphy Commission into clerical child sex abuse in the Archdiocese of Dublin.
"Following the presentation of their resignations to Pope Benedict, it has been decided that Bishop Eamonn Walsh and Bishop Raymond Field will remain as auxiliary bishops," Archbishop Diarmuid Martin said in a letter to priests of the Archdiocese given to The Irish Catholic.
The two men are to be assigned revised responsibilities within the archdiocese, according to Dr Martin.
A spokesman for the Catholic communications office said thais means the two men will be able to administer confirmation in the Dublin diocese over the next year.
Announcing their resignations in December, the two auxiliary bishops said: "It is our hope that our action may help to bring the peace and reconciliation of Jesus Christ to the victims/survivors of child sexual abuse. We again apologise to them."
Dr Walsh was appointed auxiliary bishop in Dublin in April 1990, while Dr Field was appointed on September 21st, 1997.
On November 26th 2009 the Murphy report was published. It found that allegations about 'Fr Dante' (a pseudonym) in 1997, which were also addressed by Bishop Walsh, had been dealt with appropriately by the archdiocese.
Concerning Fr Noel Reynolds, the commission report records that Bishop Walsh had been informed by a social worker that a client of hers had alleged she had been abused by Fr Reynolds. Bishop Walsh “advised her to write to the chancellor”.
The Murphy Commission found allegations against Fr Horatio (a pseudonym), and with which Bishop Field had been involved, were dealt with appropriately by the archdiocese. Where “Fr Sergius” was concerned, Bishop Field told the commission he believed he was dealing with a priest who had an alcohol problem and was not aware of abuse complaints against him.
The commission found information given by Bishop Field to priests in the parish to which 'Fr Benito' was assigned in December 2003 "was certainly not complete or sufficiently specific".
It was concerned "about the failure to inform Bishop Field about the advisory panel’s perception that he had delayed in reporting a complaint of child sex abuse".