He was ordained as deacon on 29 June 1971 by Christopher Pepys, Bishop of Buckingham at St Martin's, Fenny Stratford.

He served as Bishop of Lynn from 1994 to 1998, in which year he was appointed Dean of Windsor; in this role, Conner also held the post of Register of the Order of the Garter and was a domestic chaplain to Elizabeth II.

[7] His installation at Windsor on 7 December 1998 was attended by Elizabeth II and Prince Philip, Duke of Edinburgh;[8] he later presided over both of their interments in the King George VI Memorial Chapel on 19 September 2022, after the Queen's committal ceremony.

For many years Conner has been closely involved with the selection of candidates for ordination and with the inspection of theological colleges, courses and schemes.

[9] He was promoted to Knight Grand Cross (GCVO) on 19 July 2023 ahead of relinquishing the post of Dean of Windsor.