  The one-loop quantum corrections to geometry and thermodynamics of black hole
are studied for the two-dimensional RST model. We chose boundary conditions
corresponding to the eternal black hole being in the thermal equilibrium with
the Hawking radiation. The equations of motion are exactly integrated. The one
of the solutions obtained is the constant curvature space-time with dilaton
being a constant function. Such a solution is absent in the classical theory.
On the other hand, we derive the quantum-corrected metric (\ref{solution})
written in the Schwarzschild like form which is a deformation of the classical
black hole solution \cite{5d}. The space-time singularity occurs to be milder
than in classics and the solution admits two asymptotically flat black hole
space-times lying at "different sides" of the singularity. The thermodynamics
of the classical black hole and its quantum counterpart is formulated. The
thermodynamical quantities (energy, temperature, entropy) are calculated and
occur to be the same for both the classical and quantum-corrected black holes.
So, no quantum corrections to thermodynamics are observed. The possible
relevance of the results obtained to the four-dimensional case is discussed.
