  The semi-classical back reaction to black hole evaporation (wherein the
renormalized energy momentum tensor is taken as source of Einstein's equations)
is analyzed in detail. It is proven that the mass of a Schwarzshild black hole
decreases according to Hawking's law $dM/dt = - C/ M^2$ where $C$ is a constant
of order one and that the particles are emitted with a thermal spectrum at
temperature $1/8\pi M(t)$.
