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The Churchwardens Measure 2001 states:
There are three other requirements which a person wishing to be chosen as churchwarden must normally satisfy, but which the bishop may dispense with in exceptional circumstances.
The first of these is that a churchwarden must have his or her name on the church electoral roll of the parish. There are several reasons for this:-
(a) This requirement ensures that a churchwarden is a lay person. In view of the churchwardensβ duties as leaders of the laity [...] it is essential in all normal circumstances that they should themselves be members of the laity.
So an ordained minister could only be appointed churchwarden with the permission of the bishop, and this would only be granted in exceptional circumstances.