Rt Rev Kieran Conry, bishop of Arundel and Brighton and chair of the Bishops' Conference Department of Evangelisation and Catechesis, made the announcement in a letter read at services this weekend.
The bishop said he had been "unfaithful to my promises as a Catholic priest", but stressed his actions were not illegal and did not involve children.
His letter said: "I am sorry to confess that, going back some years, I have been unfaithful to my promises as a Catholic priest.
!I would like to reassure you that my actions were not illegal and did not involve minors. As a result, however, I have decided to offer my resignation as bishop with immediate effect and will now take some time to consider my future.
"I want to apologise to the individuals hurt by my actions and then to all of those inside and outside the diocese. I am sorry for the shame that I have brought on the diocese and the Church and I ask for your prayers and forgiveness."
It is not known what Bishop Kieran has done, but allegations have been made against him in a national newspaper.
Cardinal Vincent Nichols said: "This is a sad and painful moment. All involved in this situation are much in my prayers today."
Bishop Kieran had been a liberal in the Church and earlier this month took part in an interview with Premier, saying churches needed to be more modern and welcoming.