According to the Smart Cover page on the Apple store:
The iPad Smart Cover is compatible with iPad 2 or later.
Also at the bottom of the page:
So, the answer, based what Apple has said, is yes, the original Smart Cover will fit both the iPad 2 and the iPad 3.
There are also several questions linked in the product page that say that the cover will fit.
I did mean to update this when I found the link, but I lost it for a time and recently found it again.
According to a MacRumors report on March 22nd, some people did (and presumably still are) have some trouble with the magnets on the iPad 3. According to Marc Booth, Apple changed the polarity of the magnets sometime in 2011, because "some iPad 2 customers were having issues with their iPads entering sleep mode when they flipped Apple's Smart Cover around flat behind their iPad 2". Marc also reports that "Apple is replacing older Apple Smart Covers at Apple Stores for customers who ask."