Based on the story written by Hans Christian Andersen, I have to hand it to Disney for resisting the urge to ‘dumb’ the story down and give everyone a 100% happy ending.
Technically it is still kind of a happy ending and they made it so you could miss the fact that she is actually dead, but still; they could have handled it a lot worse.
According to comments on the video, this short was originally meant to be included in the newest Fantasia. However, it was instead it was tucked in as an added bonus on another movie.
Well, at least Disney was brave enough to put it out there.