Any vacancy in the office of President of Council, whether occurring by death, disqualification, removal or resignation, or by succession to the office of Mayor, shall be filled by the remaining members of Council, which appointee shall hold office for the balance of the unexpired term or until his successor is elected and qualified. If the vacancy be not so filled within thirty (30) days, the Mayor shall fill it by appointment. Council shall select or the Mayor appoint one of the members of Council to become President.