We study the BRST cohomology for an SL(2,R)U(1) coset model, which describes an exact string black hole solution. It is shown that the physical spectrum could contain not only the extra discrete states corresponding to those in c=1 two-dimensional gravity but also many additional new states with a ghost number NEP=-1 to 2. We also discuss characters for nonunitary representations and the relation of our results to other approaches. © 1993 The American Physical Society.