  We study the evaporation process of 2D black holes in thermal equilibrium
when the incoming radiation is turned off. Our analysis is based on two
different classes of 2D dilaton gravity models which are exactly solvable in
the semiclassical aproximation including back-reaction. We consider a one
parameter family of models interpolating between the Russo-Susskind-Thorlacius
and Bose-Parker-Peleg models. We find that the end-state geometry is the same
as the one coming from an evaporating black hole formed by gravitational
collapse. We also study the quantum evolution of black holes arising in a model
with classical action $S = {1\over2\pi} \int d^2x \sqrt{-g} (R\phi +
4\lambda^2e^{\beta\phi})$. The black hole temperature is proportional to the
mass and the exact semiclassical solution indicates that these black holes
never disappear completely.
