  We prove the Verlinde conjecture in the following general form: 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. (ii) 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.) Then the matrices formed by
the fusion rules among the irreducible V-modules are diagonalized by the matrix
given by the action of the modular transformation \tau\mapsto -1/\tau on the
space of characters of irreducible V-modules. Using this result, we obtain the
Verlinde formula for the fusion rules. We also prove that the matrix associated
to the modular transformation \tau\mapsto -1/\tau is symmetric.
