Accusations that Bell abused a young child between the late 1940s and early 1950s were made in 1995 but the Church has now admitted the response "fell a long way short".
The survivor told the then Bishop of Chichester, Eric Kemp, about the abuse in August 1995 but they were only offered pastoral support and the allegations were not referred to the police.
The Church said from the information it has, it seems Bishop Kemp did not investigate the matter further.
It was not until the claim was made to Lambeth Palace in 2013 that the survivor was put in touch with the safeguarding team at the Diocese of Chichester who referred the matter to the police and offered personal support and counselling to the survivor.
Compensation has now been agreed with the survivor, although the figure is not known.
The serving Bishop of Chichester, Rt Revd Dr Martin Warner, said he felt "deep sorrow" and acknowledged that "the abuse of children is a criminal act and a devastating betrayal of trust that should never occur in any situation, particularly the church."
He paid tribute to the survivor's courage in coming forward and said: "Along with my colleagues throughout the church, I am committed to ensuring that the past is handled with honesty and transparency."
Tracey Emmott, the solicitor for the survivor, said: "The new culture of openness in the Church of England is genuinely refreshing and seems to represent a proper recognition of the dark secrets of its past, many of which may still not have come to light.
"While my client is glad this case is over, they remain bitter that their 1995 complaint was not properly listened to or dealt with until my client made contact with Archbishop Justin Welby's office in 2013.
"That failure to respond properly was very damaging, and combined with the abuse that was suffered has had a profound effect on my client's life.
"For my client, the compensation finally received does not change anything. How could any amount of money possibly compensate for childhood abuse?
"However, my client recognises that it represents a token of apology. What mattered to my client most and has brought more closure than anything was the personal letter my client has recently received from the Bishop of Chichester."
In his letter to the survivor, Bishop Martin acknowledged that the response from the Diocese of Chichester in 1995, when the survivor first came forward, "fell a long way short, not just of what is expected now, but of what we now appreciate you should have had a right to expect then."
Sussex Police had said that if Bishop Bell had still been alive he would have been arrested and likely taken to court.
A statement from the Church said it "takes any allegations of abuse very seriously and is committed to being a safe place for all."
It added: "Any survivors or those with information about church-related abuse must always feel free to come forward knowing that they will be listened to in confidence."