South Dakota’s Republican governor, Mike Rounds, would appoint a replacement to serve until the 2008 election should Johnson die or resign. The appointment would last until the next general election — in this case, 2008. Johnson's term expires that year.
But that's not how I read the relevant South Dakota statutes. On the matter of vacancies, the law states:
Temporary appointment by Governor to fill vacancy in United States Senate. Pursuant to the Seventeenth Amendment to the Constitution of the United States of America, the Governor may fill by temporary appointment, until a special election is held pursuant to this chapter, vacancies in the office of senator in the Senate of the United States.
The law also states that the governor must call a special election within 90 days:
Special election to fill congressional vacancy--Time of election of representative. If a vacancy occurs in the office of a senator or representative in the United States Congress it shall be the duty of the Governor within ten days of the occurrence, to issue a proclamation setting the date of and calling for a special election for the purpose of filling such vacancy. If either a primary or general election is to be held within six months, an election to fill a vacancy in the office of representative in the United States Congress shall be held in conjunction with that election, otherwise the election shall be held not less than eighty nor more than ninety days after the vacancy occurs.