It seems I have to clear up some confusion. This song is called "Bella's Lullaby" by Carter Burwell. Many other people have posted with similar titles under "Bella's Lullaby" and/or Carter Burwell, but some of the videos posted are not truly Bella's official lullaby such as River Flows in You by Yiruma, which has become very popular. There happens to be another song on the soundtrack with the same title...and this song did not appear on the soundtrack. But it is still Bella's Lullaby.
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Bella's Lullaby by Carter Burwell This is the scene from Twilight where Edward plays Bella's lullaby on the piano. This IS Bella's official lullaby as this was the scene seen in the movie.