Everything I've read on the subject of what happens if Senator Tim Johnson is no longer able to serve (e.g., this and this) suggests that Mike Rounds, South Dakota's GOP governor, gets to appoint a replacement who'll serve until the end of 2008. But that doesn't seem to be what South Dakota law says: (I WAS WRONG -- SEE UPDATE BELOW)
12-11-1. 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.
If that's the case, then there'd have to be an election within three months of the declaration of a vacancy, assuming the vacancy occurs more than six months before the '08 primaries in South Dakota.
Also see the 17th Amendment to the U.S. Constitution (emphasis mine):
When vacancies happen in the representation of any State in the Senate, the executive authority of each State shall issue writs of election to fill such vacancies: Provided That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
So federal law defers to state law, therefore it looks as if there would need to be a special election -- or am I missing something?
(And I have to give credit where credit is due: I found the South Dakota statute via this Freeper.)
UPDATE: As Adrian points out in comments, the statute I quoted applies only to the House. This is for the Senate:
12-11-5. Special election to fill senate vacancy. The special election to fill the vacancy of a senator shall be held at the same time as the next general election. The general election laws shall apply unless inconsistent with this chapter.
My mistake. Thanks, Adrian.