We show that the most general two-matrix model with bilinear coupling
underlies c = 1 string theory. More precisely we prove that W-1+infini
ty constraints, a subset of the correlation functions and the integrab
le hierarchy characterizing such two-matrix model, correspond exactly
to the W-1+infinity constraints, to the discrete tachyon correlation f
unctions and to the integrable hierarchy of the c = 1 string theory.