King has stopped uprising in besieged city, private location of US siege (8)
I believe the answer is:
yorktown
'location of us siege' is the definition.
(site of 1871 siege in American Revolutionary War)
'king has stopped uprising in besieged city private' is the wordplay.
'king' becomes 'K' (abbreviation for king in chess).
'has stopped' means one lot of letters goes inside another (stop can mean to plug a gap).
'uprising in' is a reversal indicator (in a down clue, letters go up).
'besieged city' becomes 'Troy' (city besieged for ten years in Greek myth).
'private' becomes 'own' (as in one's own or private belongings).
'troy' in reverse letter order is 'yort'.
'k' put inside 'yort' is 'yorkt'.
'yorkt'+'own'='YORKTOWN'
(Other definitions for yorktown that I've seen before include "Civil War battle" , "British surrender here, 1781" , "Site of decisive seventeen hundred and eighty one British defeat in Virginia" , "US siege site" , "final victory of war" .)