c=1 String Theory as a Topological G/G Model

The physical states on the free field Fock space of the {SL(2,R)\over SL(2,R) model at any level are computed. Using a similarity transformation on $Q_{BRST}$, the cohomology of the latter is mapped into a direct sum of simpler cohomologies. We show a one to one correspondence between the states of the $k=-1$ model and those of the $c=1$ string model. A full equivalence between the {SL(2,R)\over SL(2,R) and {SL(2,R)\over U(1) models at the level of their Fock space cohomologies is found.