  Let V be a simple vertex operator algebra satisfying the following
conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0,
the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is
isomorphic to V as a V-module. Every weak V-module gradable by nonnegative
integers is completely reducible. (iii) V is C_2-cofinite. (In the presence of
Condition (i), Conditions (ii) and (iii) are equivalent to a single condition,
namely, that every weak V-module is completely reducible.) Using the results
obtained by the author in the formulation and proof of the general version of
the Verlinde conjecture and in the proof of the Verlinde formula, we prove that
the braided tensor category structure on the category of V-modules is rigid,
balanced and nondegenerate. In particular, the category of V-modules has a
natural structure of modular tensor category. We also prove that the
tensor-categorical dimension of an irreducible V-module is the reciprocal of a
suitable matrix element of the fusing isomorphism under a suitable basis.
