Defeat finally conceded -- woe is ancient besieged city (7)
I believe the answer is:
destroy
'defeat' is the definition.
(destroying is a kind of defeating)
'finally conceded woe is ancient besieged city' is the wordplay.
'finally' suggests the final letters.
'ancient besieged city' becomes 'troy' (I've seen this before).
The last letters of 'conceded woe is' are 'des'.
'des'+'troy'='DESTROY'
(Other definitions for destroy that I've seen before include "Completely ruin" , "Flatten" , "Wreck" , "'Demolish, ruin (7)'" , "Ruin, put an end to" .)