  We describe in detail how one can extract space-time geometry in an exactly
solvable model of quantum dilaton gravity of the type proposed by Callan,
Giddings, Harvey and Strominger ( CGHS ). Based on our previous work, in which
a model with 24 massless matter scalars was quantized rigorously in BRST
operator formalism, we compute, without approximation, mean values of the
matter stress-energy tensor, the inverse metric and some related quantities in
a class of coherent physical states constructed in a specific gauge within the
conformal gauge. Our states are so designed as to describe a variety of
space-time in which in-falling matter energy distribution produces a black hole
with or without naked sigularity. In particular, we have been able to produce
the prototypical configuration first discovered by CGHS, in which a ( smeared )
matter shock wave produces a black hole without naked sigularity.
