What if there is an Electoral Tie?

I was tooling around the New York Times website and they have a really cool interactive graphic for the 2004 election. While I was playing around with which states would go to Kerry and which states would go to Bush, I setup the map (quite by accident) where there was an electoral vote tie. 269-269 The map did not project a winner! What would happen? So I did some checking and this is the procedure for fixing a tie (according to Amendment 12 of the Constitution):

The electors will meet in their state capitols and cast their actual votes. These will be opened and read by the President of the Senate in a joint session of Congress. Once it has been established there is a tie, it becomes the responsibility of the House of Representatives to choose the next President of the United States. However, a special voting procedure is employed in this situation. Each state receives only one vote, and the representatives from each state have to decide which candidate gets the vote. For example, all of Indiana's representatives will vote individually. Then their votes are tallied. Which ever candidate receives a majority of these votes wins Indiana's one vote; if no candidate receives a majority of these votes, Indiana's one vote is not cast. Members from at least 34 states must be present for this vote to count. Which ever candidate receives the votes of at least 26 states is declared the President. If no one reaches 26 votes, further balloting is done. (It took the Representatives 36 ballots over 6 days to reach a winner in the 1800 election!). But that's not all. The Senate Representatives are the ones that will vote for the Vice President and their choice doesn't have to match the ticket! Which in my opinion would be way cool. Each Senator gets one vote and a minumum of 67 senators need to be present for the vote to count.

Looking at some projections by polls, this scenario, athough unlikely, is quite possible. (Note, the following sites have dynamic data so they may not match up with what I have just written)
http://electionprojection.com/elections2004.html
http://www.electoral-vote.com/