Whenever the office of a Councilman shall become vacant, the vacancy shall be filled by a majority vote of all the remaining members of the Council, and if the vacancy shall not be so filled within forty-five days, the Mayor shall fill it by appointment. Each Councilman so elected or appointed to fill a vacancy shall hold office for the balance of the unexpired term or until the beginning of the term of a successor elected for the unexpired term as hereinafter provided. A successor shall be elected for the unexpired term at the next municipal election if (a) such election shall occur more than two years prior to the expiration of the unexpired term, and (b) the vacancy shall have occurred more than one hundred twenty days prior to such election. The vacancy in the office of Councilman by reason of the adoption of this Charter shall be filled as provided in this section and for a term ending December 31, 1967.