Is it heart that's stopped in church? (5)

I believe the answer is:

organ

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'that's stopped in church?' is the definition.
(a church organ contains stops)

I can't explain the remainder of the clue.

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(Other definitions for organ that I've seen before include "Functional unit of the body and musical instrument" , "Agency" , "ear maybe" , "Wind and keyboard instrument" , "Wurlitzer, for example" .)

I've seen this clue in The Sun.

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