The problems of the existence of a competitive equilibrium in models with hidden knowledge and self-selection are well-known (Rothschild and Stiglitz (1976), Wilson (1977), Riley (1979). Recent analyses of such models argue for a particular outcome--the Pareto-dominant separating, zero-profit one (Cho and Kreps (1986), Riley (1985)). We prove the existence of such an outcome under very general conditions; and, generalizing the reactive equilibrium concept introduced by Riley (1979), we prove this outcome is the unique reactive equilibrium. This paper also introduces the strong equilibrium--the largest set of offers each of which at least breaks even on all who accept--which provides a direct and general proof of the existence of a unique reactive equilibrium.