The Governor will appoint an interim Senator for two years, and then we have an election.
Biden can pick "which" governor however.
Delaware is really "two" states - above-the-canal and below-the-canal. We are "east" of the Mason Dixon Line. North of the canal, Delaware is a typical northeastern state. Below the canal, Delaware is decidedly a southern state, except for the resort area at the beaches.
Our current governor is Ruth Ann Minner (D). She has hit her term limit, and the Democratic candidate Jack Markell is almost certain to be elected. Ruth is a downstate Democrat, Jack is an upstate Democrat. Yeah, I know, our state is 100 miles long and 5-10 miles wide - it's a little larger than the Gaza Strip.
Here's the interesting part. The federal term of election ends at
NOON on January 20th. However, the terms in Delaware start
ON January 20th.
In other words, Biden can resign his Senate seat any time up to the 20th, and Minner will appoint the interim replacement. Alternatively Biden can resign the morning of the 20th, and Markell will appoint the interim replacement, since he will be governor that morning.
Now, Biden's senate term is also subject to election this year. His opponent is a real hoot:
http://christineodonnell08.com/ - she's Sarah Palin lite.
Since Biden can control which governor picks the replacement, it is likely that a consensus has already been reached. One of the leading names tossed around is a former DE Supreme Court judge that you've never heard of, trust me.