  We investigate the proposal by Callan, Giddings, Harvey and Strominger (CGHS)
that two dimensional quantum fluctuations can eliminate the singularities and
horizons formed by matter collapsing on the nonsingular extremal black hole of
dilaton gravity. We argue that this scenario could in principle resolve all of
the paradoxes connected with Hawking evaporation of black holes. However, we
show that the generic solution of the model of CGHS is singular. We propose
modifications of their model which may allow the scenario to be realized in a
consistent manner.
