One's rather old-fashioned cul-de-sac (7)
I believe the answer is:
impasse
'cul-de-sac' is the definition.
(both can mean a stalemate)
'one's rather old-fashioned' is the wordplay.
'one's' becomes 'I'm' ('one is' is a formal way of saying 'I am').
'rather old-fashioned' becomes 'passe' (passe can mean outdated).
'im'+'passe'='IMPASSE'
(Other definitions for impasse that I've seen before include "Inescapable perdicament, dilemma" , "hopeless situation?" , "Deadlock or stand-off" , "failure to progress" , "'Deadlock, stalemate (7)'" .)