Although the letters do not disclose a sexual relationship, 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.
The Mail on Sunday has also seen a love letter from the bishop to a second woman, whose name we are not publishing for legal reasons.
The bishop insisted that the affair with the mother of two was not the reason for his resignation. He said he had quit over a different relationship.
Confronted at his home last night, 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.'
Approached by this newspaper in June, the bishop denied any sexual relationship with the married woman but admitted she had stayed at his house twice, adding: 'She is not the only woman who has stayed here.'
His statement, to be read in parishes in his diocese this weekend, from the diocesen news blog.
Bishop Kieran Conry has offered his resignation as Bishop of the Diocese of Arundel & Brighton. In a statement he said:
Statement by Bishop Kieran Conry
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.
There will be no further comment.