Scandals involving Catholic priests have been reported in other countries, including the Pope's native Germany, Switzerland, the Netherlands, Austria and the US.
Addressing the victims of abuse, the Pope wrote: "You have suffered grievously and I am truly sorry."
He continued: "Your trust has been betrayed and your dignity has been violated... I openly express the shame and remorse that we all feel." He said those guilty of abuse must "answer before God and properly constituted tribunals for the sinful and criminal actions they have committed".
David Willey, BBC News, Rome
The pope calls for healing, reparation and renewal in the Irish Church but essentially his remedy is more prayer.
He does however criticise inadequate procedures in selecting candidates for the priesthood, and the misplaced concern of some bishops to avoid scandal at all costs, even failing to report cases of paedophile priests to the police.
Victims of sexual abuse by priests in the Pope's own country, Germany, and other parts of the world are likely to be disappointed that he concentrates entirely upon the sexual scandals which occurred in Ireland without referring to the fact that this is openly acknowledged by church officials to be a worldwide problem.
However, the Pope's statement may fall short of demands by victims' groups for an admission that abuse was systematically covered up, says BBC Religious Affairs correspondent Robert Pigott.
The only direct reference to cover-ups is a reference to "a misplaced concern for the reputation of the Church and the avoidance of scandal".
The leader of one Irish victims' group said she was disappointed the Pope's letter did not recognise any responsibility of the Vatican in the crisis.
"We are really disappointed that there is no acknowledgement that the cover-up and the policy of cover-up in the church went right to the the Vatican," said Maeve Lewis, executive director of One in Four.
She said victims had also hoped for acknowledgement of the "scurrilous ways" in which they were treated as they tried to alert Church authorities to their abuse.
However, Patrick Walsh of Irish Survivors of Abuse said the letter was "unprecedented" and "encouraging".
The Pope also criticised inadequate procedures in selecting candidates for the priesthood and "insufficient human, moral, intellectual and spiritual formation in seminaries and novitiates" as factors contributing to the crisis.
He said: "Urgent action is needed to address these factors, which have had such tragic consequences in the lives of victims and their families, and have obscured the light of the Gospel to a degree that not even centuries of persecution succeeded in doing.
Vatican spokesman: "The Pope insists that abusers face the full force of justice"
Although the Pope said Vatican officials would visit Ireland to inspect some dioceses, he did not call for any restructuring of the Church in Ireland.
He also did not call for the resignation of any bishops, although a few have already volunteered to leave their posts.
The head of the Catholic Church in Ireland, Cardinal Sean Brady, has resisted calls to resign over his handling of abuse allegations in the 1970s that saw victims sign confidentiality agreements.