  We show how the two--matrix model and Toda lattice hierarchy presented in a
previous paper can be solved exactly: we obtain compact formulas for
correlators of pure tachyonic states at every genus. We then extend the model
to incorporate a set of discrete states organized in finite dimensional $sl_2$
representations. We solve also this extended model and find the correlators of
the discrete states by means of the $W$ constraints and the flow equations. Our
results coincide with the ones existing in the literature in those cases in
which particular correlators have been explicitly calculated. We conclude that
the extented two--matrix model is a realization of the discrete states of $c=1$
string theory.
