  We present a detailed analysis of $AdS_3$ gravity, the BTZ black hole and the
associated conformal field theories (CFTs). In particular we focus on the
non-extreme six-dimensional string solution with background metric $AdS_3
\times S^3$ near the horizon. In addition we introduce momentum modes along the
string, corresponding to a BTZ black hole, and a Taub-NUT soliton in the
transverse Euclidean space. We show that the $AdS_3$ space-time of this
configuration has the spatial geometry of an annulus with a Liouville model at
the outer boundary and a two-dimensional black hole at the inner boundary.
These CFTs provide the dynamical degrees of freedom of the three-dimensional
effective model and, together with the CFT corresponding to $S^3$, provide a
statistical interpretation of the corresponding Bekenstein-Hawking entropy. We
test the proposed exact black hole entropy, which should hold to all orders in
$\alpha^\prime$, by an independent field theoretical analysis including
higher-order curvature corrections. We find consistent results that yield a
renormalization of the classical parameters, only. In addition we find a
logarithmic subleading black hole entropy coming from gravitational fivebrane
instantons in a special limit in moduli space.
