A vacancy in the office of Mayor, occurring for any reason, shall be filled in the same manner as that provided for filling vacancies in Council as set out in Section 3.05 of this Charter; except Council shall not select a member of Council to fill such vacancy. However, whenever the Mayor is unable because of illness or absence from the City to perform the duties of the Mayor a Councilmember at large who shall be designated by a majority vote of Council shall perform the duties of the Mayor and be vested with all of his power and authority during such absence.