Edifice rebuilt by Abe? (5)
I believe the answer is:
abbey
'edifice' is the definition.
(abbey is a kind of edifice)
'rebuilt by abe?' is the wordplay.
'rebuilt' is an anagram indicator.
'by'+'abe'='byabe'
'byabe' is an anagram of 'ABBEY'.
(Other definitions for abbey that I've seen before include "Home for monks or nuns" , "Northanger --" , "theatre" , "Monks' abode" , "eg Bath, Fountains" .)