  The following integrability theorem for vertex operator algebras V satisfying
some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex
operator subalgebra generated by a simple Lie subalgebra {\frak g} of the
weight one subspace V_1 is isomorphic to the irreducible highest weight
\hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable
\hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie
subalgebra is also considered, and several consequences of integrability are
discussed.
