I'm reading Stephen E. Ambrose's "Undaunted Courage", and in pg. 50, he writes:
On December 5, 1800, Lewis was promoted to captain. That month the states selected their delegates to the Electoral College. In February 1801, those delegates created a political crisis when the count came out seventy-three votes each for Jefferson and his running mate, Aaron Burr, with sixty-five votes for Adams. The tie threw the election into the House of Representatives, where another deadlock followed, as the Federalist caucus decided to back Burr. In other words, the Federalists would not accept the outcome of the election, in which the people's choice of Jefferson was clear.
So intense was the partisanship of the day, so much did the Federalist hate and fear Jefferson, that they were ready to turn the country over to Aaron Burr. Had they succeeded and made Burr the president, there would almost certainly be no republic today. Fortunately for all, Hamilton was smart enough and honest enough to realize that Jefferson was the lesser evil. He used his influence to break the deadlock. On the thirty-sixth ballot, February 17, 1801, Jefferson was chosen president and Burr was elected vice-president. [Emphasis is mine]
Ambrose doesn't offer any clarification of his claim that had Burr been chosen as president, there would be no republic today. Why would Burr's presidency have led to the end of the USA?