  We study the quantum gravitational effects in spherically symmetric black
hole spacetimes. The effective quantum spacetime felt by a point-like test mass
is constructed by ``renormalization group improving'' the Schwarzschild metric.
The key ingredient is the running Newton constant which is obtained from the
exact evolution equation for the effective average action. The conformal
structure of the quantum spacetime depends on its ADM-mass M and it is similar
to that of the classical Reissner-Nordstrom black hole. For M larger than,
equal to, and smaller than a certain critical mass $M_{\rm cr}$ the spacetime
has two, one and no horizon(s), respectively. Its Hawking temperature, specific
heat capacity and entropy are computed as a function of M. It is argued that
the black hole evaporation stops when M approaches $M_{\rm cr}$ which is of the
order of the Planck mass. In this manner a ``cold'' soliton-like remnant with
the near-horizon geometry of $AdS_2\times S^2$ is formed. As a consequence of
the quantum effects, the classical singularity at r=0 is either removed
completely or it is at least much milder than classically; in the first case
the quantum spacetime has a smooth de Sitter core which would be in accord with
the cosmic censorship hypothesis even if $M<M_{\rm cr}$.
