That is an excellent question. I would have to guess that the VP elect will take over.
It's an excellent answer to a question, but not the question JLM asked.
JLM, the answer is "there is no national standard for what would happen." The Constitution says "Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress. . ." U.S. Const., Art. II, sec. 1, cl. 2.
This means that the electors are chosen as directed by state statute, and the electors choose the next President. In practice, the electors follow the popular vote in their states, but that can be changed at any time by the state legislature.
The question Smacky answered would be "What happens if the President-elect dies before taking office?" in which case his answer is correct.
See U.S. Const., Amdt XX, sec. 3.