Priest in charge of the St Michael church, Father Jeffrey Gibson, said that just before 3 p.m., the tree collapsed.
He said the church has already put plans in place to have it removed, the walkway cleared and the grave repaired.
Gibson said "the tree will be put to good use, as St Leonard's mahogany is the best mahogany around".
The minister said last Friday morning another mahogany tree on the other side of the graveyard also came down.
No one was injured and no damage was recorded.