A member of the public who stumbled across the man called emergency services but he was pronounced dead at the scene by ambulance staff.
Police remain on scene at the church in Friar Street after being called to the unexplained death at 10.05am today.
The body has yet to be identified and thought the death is being treated as unexplained it is not known whether it is suspicious at this time.
The graveyard is currently closed to pedestrians while the crumbling walls of the cemetery are reinforced.
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