We find the discrete states of the c = 1 string in the light-cone gauge of Polyakov. When the state space of the gravitational sector of the theory is taken to be the irreducible representations of the SL (2, R) current algebra, the cohomology of the theory is not the same as that in the conformal gauge. In particular, states with ghost numbers up to four appear. However, after taking the space of the theory to be the Fock space of the Wakimoto free-field representation of the SL (2, R), the light-cone and conformal gauges are equivalent. This supports the contention that the discrete states of the theory are physical. We point out that the natural states in the theory do not satisfy the KPZ constraints.