We study the effect of ZZ instantons in c = 1 string theory, and demonstrate that they give rise to non-perturbative corrections to closed string scattering amplitudes that do not saturate unitarity within the closed string sector. Beyond the leading non-perturbative order, logarithmic divergences are canceled between worldsheet diagrams of different topologies, due to the Fischler-Susskind-Polchinski mechanism. These results also allow us to propose the exact non-perturbative matrix model dual of c=1 string theory.