Is it true that a mosquito is really attracted to one's breath, or is this just one of those Internet facts that someone dreamed up because they were bored?

Yes it's true.
They're attracted to the carbon dioxide in our exhalations.

It would appear that they are! (It's news to me too!)
Here are some sites that discuss the subject:

http://www.mosquito.com.au/mosquito/facts.html

http://www.pulseplanet.com/archive/Jul00/2193.html

Minnesota and Texan-In-Exile, thanks for the great answers!