Hi Jonas,
This is probably not needed any more but just to help people, I think I found a way of doing this:
I went into the Assign roles of the Site admin block > Assigned all my students as a guest on this block (just so I can edit the permissions without affecting anyone) > then went to override permissions on the block and clicked prohibit for View Block for the guest user.
This allowed the Admin to still view the block as normal but stopped the students from seeing the block at site level without having to hide anything.
Thanks
Ant