2006 New York Code - Vacancies.

 
    § 15. Vacancies.   If   a  vacancy  shall  occur,  otherwise  than  by
  expiration of term, in an elective office of the city, including that of
  supervisor, the mayor shall appoint a person to fill such  vacancy.  The
  person  so  appointed  to  such  vacancy  shall  hold  office  until the
  commencement of the political year  next  succeeding  the  first  annual
  election  after the happening of the vacancy at which a successor can be
  elected for the balance of the unexpired term,  if  any.  If  a  vacancy
  shall  occur  in  an  appointive  office  of the city, otherwise than by
  expiration of term, the officer, officers, board or body  authorized  to
  make appointment to such office for the full term shall appoint a person
  to fill such vacancy for the balance of the unexpired term.