Ah, this. This has long been accepted as fact among a pretty large group of anthropologists. I know when I was an anth major, I firmly believed in it - the only other possibility being that Neanderthals were killed out by our ancestors.
A really good piece of evidence, beyond this DNA thing, would be the skulls. If you compare a neanderthal skull (which, by the way, has a larger brain case than our own - Neanderthals had larger brains!) to a modern Homo Sapiens, there is very little to set them apart beyond a "brow ridge" and a bun at the base of the skull.
IN fact, if you compare a modern homo sapiens skull to an early homo sapiens skull, there is more of a difference between the two than there is between sapiens and neanderthalensis!
(Some anthropologists suggest naming Neanderthals Homo Sapiens Neanderthalensis, making our species Homo Sapiens Sapiens).
Finally, a fun little piece of anatomy. Neanderthals have something at the base of their skull, where the neck meets the skull - it's a small bony protrustion. Archaic homo sapiens skulls that have been found completely lack this protrustion. However, a fairly large minority of modern homo sapiens DO have this bun - further "proof" towards this theory.
Seriously, check it out. Feel the base of your skull, and see if there's a bony protuberance - a small "nub" or "bump". I know I have one, and many people of northern european origin do as well - which makes sense, since there was a huge concentration of Neanderthals in that portion of the world