Yes, but if it breaks near the "middle" (at the point of highest stress), it'll have time to fall several thousand miles in vacuum, building up a *lot* of velocity and energy, before it encounters any atmosphere. Plus there are interesting forces similar to a game of "crack the whip" as it starts to wrap itself around the Earth.
There's a good description of the consequences of this in the science fiction book "Red Mars" (by Kim Stanley Robinson). The falling space elevator (in the book, on Mars and not on Earth) does some counterintuitive but spectacular things as it wraps around the planet 3-4 times.
Without seeing the book yet, wouldn't it basically vaporize itself upon hitting the air, freeing the remainder of the loose end which would fly out into space?