The rest of that story is that West Virginia Governor Joe Manchin, whose job it is to appoint Byrd's successor, has long aspired to that very seat, has explicitly said he wouldn't appoint himself to it. Apparently, timing's an issue right now, too. If Sen. Byrd had survived until July 3rd, it would be too late to hold a special election this fall, and whoever's appointed to replace him would serve until the 2012 election. Since he died this morning, the appointee will serve until a special election this November -- and of course could run to fill the rest of the term.
There's speculation that Manchin should go ahead and appoint himself, and run in the special election. There's no better time than right now for him to step into that seat, and if he waits to run in 2012, he might find himself facing a popular incumbent, depending on how good a job he did appointing someone now.
[Update at 5:41: The WV Secretary of State says it's already too late to hold a special election this fall, so Manchin's appointment will serve out the term, which ends January 2013. The seat will next be elected in November 2012.]