The Catholic Bishop of Arundel and Brighton has resigned after close relationships with parishioners, including one with a married woman.

Kieran Conry, head of evangelisation for the Catholic Bishops of England and Wales and the church's former head of media, said in a statement read out in all churches in his diocese at Mass this weekend, 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 first of all to the individuals hurt by my actions and then to all of those inside and outside the diocese who will be shocked, hurt and saddened to hear this.

"I am sorry for the shame that I have brought on the diocese and the Church and I ask for your prayers and forgiveness."

After the statement appeared on his website, detailed allegations were published by the Mail on Sunday in which it became clear the newspaper had been investigating him for several months and that the priestly vow the bishop admits to having broken is that of chastity.

The Mail on Sunday reports the resignation today and discloses that Bishop Conry has had a close friendship with a married mother of two. The Bishop denies that this was a sexual relationship. He has resigned over a separate relationship that he had six years ago.

Regarding the married woman, Adam Luck and Jonathan Petre report: "This newspaper has seen love letters between Bishop Conry, 63, and the woman, written in recent months, in which he tells her that her husband doesn't love her."

One handwritten letter, which was dated earlier this month and addressed "Dear, Dear XXXXX" and signed "your K xxxxxx", the Bishop said: "... It's all right to say that [your husband] did bad things, but you knew that he didn't love you. You know (I hope) that I did. And I did, and do."

Although the letters do not disclose a sexual relationship, the newspaper says, the affair lasted more than a year and involved the married woman spending at least three nights at the bishop's detached property in Pease Pottage, West Sussex.

Bishop Conry insisted to the newspaper that the affair with the mother of two was not the reason for his resignation and said he had quit over a different relationship.

Confronted at his home, he said: "This has nothing to do with your enquiries. It is totally unrelated. This relates to a relationship of six years ago. So be careful what you write. This had nothing to do with your earlier questions." When he was approached by the Mail on Sunday in June, he denied any sexual relationship.

The paper also disclosed that the woman's estranged husband employed a private detective to track her movements and is considering a legal action against the Church.

Leading Catholic commentator Damian Thompson wrote on his Spectator blog: "The Mail has now published its allegations against Bishop Conry. They're much more serious than I imagined. He appears to have behaved disgracefully; by his own admission there is more than one woman involved." Dr Thompson questioned "the responsibility of the Catholic Church for allowing this man to run an important diocese during years of rumours, well-founded in at least two cases".

He added: "I liked Kieran a lot when he was press officer for the English bishops: he didn't give us any bullshit and let slip the odd bit of gossip. We we all pleased when he was made a bishop. Sure, he was a liberal, but they all were.

"Then, shortly after taking office, he started bullying the parish of the Sacred Heart, Hove, for using too much Latin in its main Sunday Mass. I rang him up and said: this is outrageous – it's not some elaborate Old Rite service, just a dignified Novus Ordo said mostly in English with nice music (such a rarity). The distress Conry caused by barging in and trying to wreck it was enormous and completely avoidable. Since then traditionalist priests in his diocese have had their collars felt for worshipping in the style of Benedict XVI. Kieran would often refuse to answer letters from his clergy; but left-wing laity slagging off traditionalists always got a sympathetic hearing."