This would be my understanding. Widow Carnahan was appointed by the governor as happens when a Senator meets an untimely end during his/her term. The fact that she was appointed to fill the spot was probably suspect in the first place because the person elected, Mel Carnahan, was not a viable candidate at the time of the election.
Ashcroft was gentlemanly enough to acquiesce. A good case could have been made for a special election then and there.
When a Senator is appointed by the governor, I believe the Constitution requires that the seat be contested during the next election cycle. That's what's happening now. And if Thune wins, yes, he would be installed in that seat immediately--not in 4 years.
Sweet!