Whenever the office of a councilmember shall become vacant, Council shall declare the vacancy to occur or to have occurred on a certain date, and shall fill the vacancy by appointment by a majority vote of all the remaining members of Council. If the vacancy shall not be filled by Council within thirty days of the vacancy, the mayor shall fill such vacancy by appointment. Such appointee shall hold office for the balance of the unexpired term of the member in whose office the vacancy occurs or until the beginning of the term of a successor elected as hereinafter provided. A successor shall be elected to serve the remainder of the unexpired term of the member in whose office the vacancy occurs at the next regular 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.