Robert Pattinson is the singer.

The song's name is "Let Me Sign". He also composed the other song "Never Think" which is used in the scene at the Cullen's home at the piano.

He is also the actor who plays Edward Cullen.

In the movie, I believe it is Robert Pattinson's voice singing in the background.  He is featured on the soundtrack of the movie, and can also be heard singing in the background when Edward and Bella are visiting in the Italian restaurant after he rescues her from the thugs who tried to attack her in Port Angeles.

When Edward is sucking the venom out of Bella, in Chapter 23, there are only two other people present, Alice and Carlyle. As Bella drifts in her "poisoned" state, she says she heard "the happiest sound my mind could conjure up..."(452). She refers to this sound as "an angel calling my name..."(452). But as this section continues, it seems that the angel is Edward calling to Bella.  There is no mention in the chapter, at least none I could find, to anyone "singing" to Bella as Edward saves her.  Both Alice and Carlyle have some dialogue in the chapter, but there is no singing.