The piece of music that plays during the portrait scene was never meant to be in the film. Composer James Horner recorded himself playing the main theme of the film on the piano and sent it to director James Cameron titled "Sketch." Horner meant that the song was a sketch of a piece of music that he was going to flesh out later with an orchestra. Cameron interpreted the title "Sketch" to mean that it was meant for the scene where Jack sketches Rose. Cameron loved how simple and beautiful it was and decided to use it in the final film. Personally, I'm very happy it's in the movie. It's my favorite song from the soundtrack.
Isn't it amazing?