(A) A vacancy in the membership of the Council shall be filled by a majority vote of the members of the Council within 30 days after the vacancy occurs; provided, however, that Council may, by a majority vote of the members of the Council, extend this period by an additional 30 days. After the applicable 30-day or 60-day period, the power of the Council to fill the vacancy shall lapse, and the Mayor shall fill the vacancy by appointment.
(B) If the vacancy occurs on or after July 1 of the second year of a Council member's term, the person selected by the Council or appointed by the Mayor to fill such vacancy shall serve for the remainder of that unexpired term. If the vacancy occurs on or before June 30 of the second year of a Council member's term, the person selected by the Council or appointed by the Mayor shall serve until a successor is elected at the next November election. The person so elected in November shall take office on January 1 following such election and serve the remainder of the original Council term.