We study the quantum gravitational effects in spherically symmetric black h
ole spacetimes. The effective quantum spacetime felt by a pointlike test ma
ss is constructed by "renormalization group improving" the Schwarzschild me
tric. The key ingredient is the running Newton constant which is obtained f
rom the exact evolution equation for the effective average action. The conf
ormal structure of the quantum spacetime depends on its ADM mass M and it i
s similar to that of the classical Reissner-Nordstrom black hole. For M lar
ger than, equal to, and smaller than a certain critical mass M-cr the space
time has two, one, and no horizon(s), respectively. Its Hawking temperature
, specific heat capacity, and entropy are computed as a function of M. It i
s argued that the black hole evaporation stops when M approaches M-cr which
is of the order of the Planck mass. In this manner a "cold" soliton-like r
emnant with the near-horizon geometry of AdS(2) x S-2 is formed. As a conse
quence of the quantum effects, the classical singularity at r=0 is either r
emoved completely or it is at least much milder than classically; in the fi
rst case the quantum spacetime has a smooth de Sitter core which would be i
n accord with the cosmic censorship hypothesis even if M<M-cr.