Rev Rupert Kalus, of Lanchester and Burnhope Benefice, and the Rev Ali Miller were both married when they started the affair, which has now ended, a Diocese of Durham spokesman said.
The Bishop of Durham, Rt Rev Paul Butler (pictured below), broke the news to parishioners at a service on Sunday.
Following an investigation by the Church of England, both the vicar and the curate have been suspended from public ministry.
Mr Kalus was suspended for 18 months and Ms Miller for twelve months.
A Diocese of Durham spokesman described the affair as "adulterous" and said the organisation was "deeply saddened" by events.
He said: "Following the procedures of the Clergy Discipline Measure, the Bishop of Durham has prohibited both Rupert and Ali from their public ministry for a period of 18 and 12 months respectively.
"They have both therefore had to resign their posts.
"The Rt Rev Paul Butler, Bishop of Durham, confirmed these facts to the congregation at the normal Sunday service which he led on Sunday 12th August.
"We are deeply saddened that this situation has arisen and ask that you respect the privacy of both families and hold them in your prayers."
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