  Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie
algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify
irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is
completely reducible. In the special case n=1, we study a category of weak
L(-l+3/2,0)-modules which are in the category $\cal{O}$ as modules for the
associated affine Lie algebra. We classify irreducible objects in that category
and prove semisimplicity of that category.
