  The string equations of motion and constraints are solved near the horizon
and near the singularity of a Schwarzschild black hole. In a conformal gauge
such that $\tau = 0$ ($\tau$ = worldsheet time coordinate) corresponds to the
horizon ($r=1$) or to the black hole singularity ($r=0$), the string
coordinates express in power series in $\tau$ near the horizon and in power
series in $\tau^{1/5}$ around $r=0$. We compute the string invariant size and
the string energy-momentum tensor. Near the horizon both are finite and
analytic. Near the black hole singularity, the string size, the string energy
and the transverse pressures (in the angular directions) tend to infinity as
$r^{-1}$. To leading order near $r=0$, the string behaves as two dimensional
radiation. This two spatial dimensions are describing the $S^2$ sphere in the
Schwarzschild manifold.
