If a vacancy occurs in the representation of this state in the Senate of the United States, the Governor may appoint and commission an elector of this state who possesses the qualifications for the office to fill the vacancy until his or her successor is elected and qualifies and is admitted to his or her seat by the United States Senate. However, whenever a vacancy occurs within a term fixed by law to expire on the third day of January following the next general election, the person so appointed shall hold office for the remainder of the unexpired term unless the vacancy is filled at a special election held prior to the general election, in which case the person elected at the special election shall hold office for the remainder of the unexpired term. An election to fill a vacancy in the term of a United States Senator shall be held at the general election next succeeding the occurrence of the vacancy or at any special election.