The Senate meets and chooses Vicepresident. The result is 50-50. The President of the Senate breaks the split... voting for himself.
An interesting, if not chilling, scenario (you need to sell the screen rights to Hollywood). I'm not sure, however, if the vice-president, as president of the senate, would be allowed to cast a vote in this situation. The 12th Amendment states:
"...if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and [b]a majority of the whole number shall be necessary to a choice[/b]."
That last phrase can be read two ways: (1) that a candidate must receive a majority vote of senators
; or (2) that candidate must receive a majority vote of the senate
. In the first case, the vice-president would not be allowed to vote; in the second case, he would (or, at least, he'd have a much better argument for being allowed to vote).
Unfortunately, history offers us no guidance. The senate has used this procedure only once, to elect Richard Johnson
in 1837. The senate vote in that case was 33-16, so there was never any question of the vice-president casting a tie-breaking vote (ironically, the vice-president at the time was Martin Van Buren, who was also the president-elect). Strangely enough, the only time that the 12th Amendment was used to break a tie in a presidential election -- 1825 -- there was no need for the senate to elect a vice-president: John C. Calhoun was elected to that position with a clear majority of the electoral college.