The song NOT included in the theatrical release of the film, "If I Never Knew You," though put into the release of the 10th Anniversary Edition DVD.
I love this song, and I am SO glad they put it back in. Makes the whole scene much more effective. Also, this is my first Youtube upload, soo...Yay me! And if you notice the audio to be a little crackly, I do apologize.
[EDIT] To answer some of the questions in the comments, this song was originally intended for the film and was almost finished, when Disney ran the movie (with the song) to a few test audience, and the younger--read, the kid--audience pretty much started squirming during this song, probably wanting to get back to the action. However, the song performed well with the teen and adult audiences. (Taken from the SE DVD's commentary.)
Also, YES, that is Mel Gibson singing. He also does all of the rest of John Smith's singing in the movie. Surprising, I know. If you doubt, just listen to the speaking voice. Same vocal quality. Sure, Judy Kuhn sings the pants off him, but he's not bad.