An exact conformal field theory describing a black hole in two-dimensional space-time is found as an SL(2,openR)/U(1) gauged Wess-Zumino-Witten model. For k=9/4, the conformal field theory can be regarded as a classical solution of the same system that is probed in the c=1 matrix model. The conformal field theory governing the space-time is regular at the Riemannian singularity, but it appears that generic perturbations blow up there. It is argued that the end point of the Hawking black-hole evaporation is the standard space-time of the c=1 matrix model, which should be regarded as an analog of the extreme Reissner-Nordström black hole of four-dimensional general relativity. The c=1 model is thus a model of the quantum mechanics of matter interacting with a black hole.