In the letter, to be read out in Sunday Mass, the Pope said he shared the sense of dismay and betrayal felt by the Irish people over the abuse scandal.
He said the Church's handling of the abuse had been "inadequate" and that "mistakes" had been made.
He admitted that child abuse by clergy had "contributed in no small measure" to the weakening of faith and the loss of respect for the Church and its teachings.
Addressing the victims of abuse and their families, the Pope said: "You have suffered grievously and I am truly sorry. I know that nothing can undo the wrong you have endured. Your trust has been betrayed and your dignity has been violated.
"It is understandable that you find it hard to forgive or be reconciled with the Church. In her name, I openly express the shame and remorse that we all feel."
The Pope appealed to victims to find hope and healing in knowing that Jesus was also a victim of sin and close to them in their suffering.
Addressing bishops, he wrote: "It cannot be denied that some of you and your predecessors failed, at times grievously, to apply the long-established norms of canon law to the crime of child abuse. Serious mistakes were made in responding to allegations ... All this has seriously undermined your credibility and effectiveness."
The Pope proposed a path of healing, renewal and reparation within the Church and called on Irish Catholics to address the problem of abuse with "courage and determination".
"No one imagines that this painful situation will be resolved swiftly. Real progress has been made, yet much more remains to be done," he said. "Perseverance and prayer are needed, with great trust in the healing power of God's grace."
He continued: "At the same time, I must also express my conviction that, in order to recover from this grievous wound, the Church in Ireland must first acknowledge before the Lord and before others the serious sins committed against defenceless children.
"Such an acknowledgement, accompanied by sincere sorrow for the damage caused to these victims and their families, must lead to a concerted effort to ensure the protection of children from similar crimes in the future."
He said "urgent action" was required to address the scandal, including improving the spiritual preparation of candidates for the priesthood and ending the "misplaced concern for the reputation of the Church and the avoidance of scandal".
The letter is part of the Church's response to the damning Murphy and Ryan reports last year, which exposed decades of unreported sexual abuse at the hands of clergy and Church-run institutions.
The reports led to the resignations of a number of bishops who were criticised for their failure to report the abuse to the authorities.
Catholic leaders are hoping the letter will allow the Church to turn a corner. The letter was welcomed by Cardinal Sean Brady, head of the Catholic Church in Ireland.
He told the Catholic faithful, "I ask you that you read this letter with an open heart and in a spirit of faith."
The pastoral letter will have significance beyond Ireland in light of recent abuse scandals in Germany, the Netherlands, Austria and Switzerland.