  Using the conformal invariance of the $SL(2,R)\otimes SO(1,1)^{d-2}/SO(1,1)$
coset models we calculate the conformally exact metric and dilaton, to all
orders in the $1/k$ expansion. We consider both vector and axial gauging. We
find that these cosets represent two different space--time geometries: ($2d$
black hole)$\otimes \IR^{d-2}$ for the vector gauging and ($3d$ black
string)$\otimes \IR^{d-3}$ for the axial one. In particular for $d=3$ and for
the axial gauging one obtains the exact metric and dilaton of the charged black
string model introduced by Horne and Horowitz. If the value of $k$ is finite we
find two curvature singularities which degenerate to one in the semi--classical
$k\to \infty$ limit. We also calculate the reflection and transmission
coefficients for the scattering of a tachyon wave and using the Bogoliubov
transformation we find the Hawking temperature.
