Amends Sections 13.2 and 13.3 of the Charter of the City of Durham, SL 1975-671, as amended. Authorizes the City Council to appoint a qualified person to fill a vacancy occurring in the office of Mayor or a member of City Council. Requires the City Council to adopt a resolution calling a special election to fill the vacancy if the City Council chooses not to appoint an individual to fill such vacancy and chooses to call a special election (current law requires the City Council to choose a person within 60 days of the vacancy, and if the Council fails to do so within that time period, the Council is barred from choosing and must call a special election by adopting a resolution within seven days of the 60-day period).