Hi Emily,
I agree. Br- is a very weak base. But that protonated species REALLY wants to get rid of the excess proton, so it needs to go somewhere. It will either go to the Br- or to the solvent molecule. If the solvent was an alcohol, it could still deprotonate the OH2+.
It seems weird to produce a strong acid as a product, but it does happen.
KEEP THIS IN MIND: If the solvent isn't water, then HBr is just another molecule. HBr is only an acid when it DISSOLVES AND DISSOCIATES IN WATER. Sneaky sneaky!