  A renormalizable theory of quantum gravity coupled to a dilaton and conformal
matter in two space-time dimensions is analyzed. The theory is shown to be
exactly solvable classically. Included among the exact classical solutions are
configurations describing the formation of a black hole by collapsing matter.
The problem of Hawking radiation and backreaction of the metric is analyzed to
leading order in a $1/N$ expansion, where $N$ is the number of matter fields.
The results suggest that the collapsing matter radiates away all of its energy
before an event horizon has a chance to form, and black holes thereby disappear
from the quantum mechanical spectrum. It is argued that the matter
asymptotically approaches a zero-energy ``bound state'' which can carry global
quantum numbers and that a unitary $S$-matrix including such states should
exist.
