California Code, Government Code - GOV § 1752

(a) Except as provided in subdivision (b), no person elected or appointed to the governing body of any city, county, or district having an elected governing body, shall be appointed to fill any vacancy on that governing body during the term for which he or she was elected or appointed.

(b) With respect to a general law city, if a vacancy in the elected office of mayor occurs, the council may fill that vacancy by appointing a member of the council to the office of mayor.  Any person appointed to fill the vacancy shall hold the office of mayor for the unexpired term of the former mayor.  Any vacancy created in the membership of a city council as the result of an appointment of a member to the office of mayor shall be filled in accordance with Section 36512.