Dry old PM (not North) quit (6)
I believe the answer is:
secede
'quit' is the definition.
(thesaurus)
'dry old pm not north' is the wordplay.
'dry' becomes 'sec' (similar in meaning).
'old pm' becomes 'eden' (I've seen this before).
'not' indicates named letters should be taken away.
'north' becomes 'n' (synonyms).
'eden' with 'n' taken out is 'ede'.
'sec'+'ede'='SECEDE'
(Other definitions for secede that I've seen before include "'Withdraw, break away (6)'" , "Formally withdraw (from federal union)" , "Make a formal withdrawal" , "Withdraw (politically)" , "Withdraw from political union" .)