How do I treat black spot on my silver birch sapling?

Sounds like either a fungal infection or a bacterial infection. If the tree is small, you can buy a fungicide (make sure it doesn't say you can't use it on Birch) and spray the tree. Pick off really badly affected leaves and dispose of them. As you have sandy soil, you should take extra care to keep the tree watered, particularly for its first two years, so at least a gallon a week, or twice a week. It wouldn't hurt to stir in a bit of Growmore, or you could water on Phostrogen, but not past the middle of July. Birch doesn't like root disturbance, so be careful not to stir the soil too far down.

Thank you very much, that's really helpful. I'll try all those things!