Two-matrix model and c=1 string theory

We show that the most general two--matrix model with bilinear coupling underlies $c=1$ string theory. More precisely we prove that $W_{1+\infty}$ constraints, a subset of the correlation functions and the integrable hierarchy characterizing such two--matrix model, correspond exactly to the $W_{1+\infty}$ constraints, to the discrete tachyon correlation functions and to the integrable hierarchy of the $c=1$ string.