Vacancies in the office of Councilman shall be filled within sixty (60) days by vote of a majority of the remaining Council members by the selection of a person qualified as provided in § 4.03 hereof. If the Council should fail to act within sixty (60) days, the Mayor, or Deputy-Mayor when acting as Mayor, shall fill such vacancy by appointment. A person so chosen or appointed shall serve until the next regular municipal election occurring not less than one hundred (100) days after his selection. At such election a successor shall be elected to serve for the remainder of the unexpired term, if any; if not, for a full term.