Courtesy of Ishan Shah:
Municipal Code, Title II, Chapter I (City Council & Mayor), Section 2-1106: Filling Council Vacancies.
This section provides for the following:
(a) If a vacancy occurs in an office of councilmember or in the office of mayor, the city council shall, within 30 days from the commencement of the vacancy, fill the vacancy in accord with the provisions of this section.
(b) The city council may appoint a person to fill the vacancy, to hold office until the date of the certification of the results of a special election which will immediately be called to be held on, either the date of the next regularly established election or on the date of the next regularly scheduled municipal election, to be held throughout the city and not less than 90 days from the call of the special election.
(c) If the city council fails to appoint a person to fill the vacancy, the city council shall immediately call a special election to be held on the date of the next regularly established election.
(d) Notwithstanding subsections (b) and (c), a special election called to fill a vacancy caused by a recall of a councilmember or mayor shall be called in accordance with Elections Code Section 27333.