There isn't a song with just Edward and Bella, at least not that I know of.
One of the most popular songs from the Twilight Saga, "Flightless Bird, American Mouth" by Iron & Wine, features a scene with just Edward and Bella at the prom in the first movie. It is a slow, romantic song that captures the emotions of the characters in that moment.
Bella never sings to Edward.
the song that Edward composed himself for Bella.
Bella's lullaby
In the Twilight book series by Stephenie Meyer, Bella Swan's favorite song is "Flightless Bird, American Mouth" by Iron & Wine.
Bella's Lullaby by Carter Burwell
The lion fell in love with the lamb?
Bellas lullaby
Edward gave Bella a CD with the song he wrote for her recorded in it.
if you are talking about the song that edward plays for Bella it is called Bella's Lullaby and it is by Carter Burwell
the song that edward plays for Bella in twilight is called Bella's lullaby. he wrote it for her when they first met. its beautiful!!! actually there are several "Bella's lullabys" on this album..the name of the actual song is "kiss the rain" by yiruma the other "Bella's lullaby" is by carter burwell
the song that was playing while edward sucked the venom out of Bella's arm where James had bitten her was "LET ME SIGN"
Yes, The song will actually be "River Flows In You" by Yurima. You can hear it on playlist.com, just search for "Edward Cullen" =)