The following questions and answers might also interest you.

"What happens if the candidate who wins on Election Day dies or resigns before Inauguration Day?
One possible scenario is if the candidate dies or resigns in between the popular vote and the Electoral College vote. Both major political parties have rules that allow them to replace their candidates on the ballot after the party convention. If, for example, Republican X won a majority of pledged electoral votes on Election Day but died the week before the Electoral College met, the Republican Party would tell the electors pledged to Republican X to vote for replacement Republican Y. (There is no legal problem with this replacement, but the party is unable to force its pledged electors to vote a certain way. Whether or not the electors would follow the national party’s instructions is unclear.)

The most difficult scenario occurs should the winning candidate die or resign between the meeting of the Electoral College in mid-December and the counting of the electoral votes by Congress on January 6. If the Democratic Candidate X received a majority of the Electoral College votes when the electors met in mid-December, but dies before the Congress officially counts the votes from the states, Congress has two options:

The first option is to count all of the votes as received from the states. In our example, Democratic Candidate X received a majority of the votes and would be declared the president-elect of the United States. On Inauguration Day, the provisions of the Presidential Succession Act of 1947 would go into effect and the vice president-elect would become the president of the United States. The second option is for Congress to invalidate all of the electoral votes for the deceased candidate. In this instance, there is likely to be a vice president-elect but no candidate with a majority of votes for presidency. The presidential contest would be decided by the House of Representatives where the state delegations would only be able to consider the other candidate(s) receiving electoral votes.

The final scenario is if the president-elect dies or resigns after the electoral vote is counted by Congress but before Inauguration Day. The Presidential Succession Act of 1947 would go into effect on Inauguration Day."

Of note is that in your scenario, yes the VP would take over, but because of the Presidential Succession Act of 1947 and not because of the Constitution. This also answers what happens if the VP then dies, etc.
Source(s):
http://www.eac.gov/voter/electoral-college-faqs
http://en.wikipedia.org/wiki/Presidential_Succession_Act

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