Old bird in repeated act (4)
I believe the answer is:
dodo
'old bird' is the definition.
(extinct bird)
'repeated act' is the wordplay.
'repeated' says to double the letters.
'act' becomes 'do' (to do is to act or execute a task).
'do' duplicated is 'DODO'.
'in' acts as a link.
(Other definitions for dodo that I've seen before include "Two rooks - one was unable to fly" , "Old bird" , "one finished?" , "one very dead!" , "Proverbially extinct bird" .)