Spoiler --
I saw up to the point where she reads him the actual letter. They hug, and.... the library DVD freezes irrevocably. I can't play anything after that. It's all big pixels and freezing again.
Could anyone type out a few key lines to tell me how it ends?
Well, probably not reading anymore, but they part. She goes home and talks to her mom about what they will do for her birthday. Seems they will celebrate it at Tivoli. For the first time in the movie she really does seem like she's 14.
He takes a train to Skagen and meets his grandfather, who is completely uninterested and generally a jerk, so now we see how his father and he turned out that way.
Finally he goes to the spit and sees where the two seas -- Baltic and North -- run into one another as the woman had described earlier, and that's it.