Any vacancy in Council shall be filled by a majority vote of the remaining members of Council. If such vacancy is not filled within sixty days after it occurs, then the Mayor may fill it by appointment made by a written document filed with the Clerk of Council. Council's authority to fill a vacancy shall not terminate at the end of said sixty day period, thereafter, the Council and Mayor shall have concurrent authority to fill such vacancy. Council, after such sixty day period, may fill such vacancy at any regular meeting held prior to the filing with the Council of a written appointment by the Mayor filling such vacancy. Any person selected to fill a vacancy in Council shall serve for the unexpired term unless the vacancy occurs prior to July 1 in the second year of said term, in which event an election shall be held at the next scheduled general municipal election at which members of Council are elected to fill the balance of the unexpired term, and the term of the person appointed by the Council shall terminate upon the certification of election of a person to fill the balance of the unexpired term.