Sussex Police accepted it should have informed the deceased clergyman's relatives before confirming it would have charged him with sex abuse had he still been alive.
In October 2015, the Church accepted assault allegations against Rt Revd George Bell, who was Bishop of Chichester from 1929 until his death on 3rd October 1958, and paid compensation to the alleged survivor.
In a statement, officials said there was no "reason to doubt the veracity of the claim" and apologised for his actions.
The way the case has been handled by the Church and police has been under scrutiny since, with some saying it was wrong to vilify a man who was not alive to defend himself.
Journalist Peter Hitchens was a high profile voice condemning the allegations being made public without the prospect of a fair trial.
He wrote a letter of complaint to Sussex Police. In the force's reply it said an apology would be made to Bishop Bell's niece Barbara Whitley.
Detective Superintendent Jeremy Graves, is quoted by the BBC as writing: "The distress caused to Barbara Whitley is of course regrettable and I know that Katie Perkin [head of corporate communications] plans to personally write a letter of apology to her.
"With hindsight the matter could have been managed more sensitively but it was complicated by the fact that the release was generated by the diocese with whom we should have been working more closely.
"I am satisfied there was no intention to confuse or cause distress."
The Church of England has been contacted for a comment.
