  The string propagation in the two-dimensional stringy black-hole is
investigated from a new approach. We completely solve the classical and quantum
string dynamics in the lorentzian and euclidean regimes. In the lorentzian case
all the physics reduces to a massless scalar particle described by a
Klein-Gordon type equation with a singular effective potential. The scattering
matrix is found and it reproduces the results obtained by coset CFT techniques.
It factorizes into two pieces : an elastic coulombian amplitude and an
absorption part. In both parts, an infinite sequence of imaginary poles in the
energy appear. The generic features of string propagation in curved
D-dimensional backgrounds (string stretching, fall into spacetime
singularities) are analyzed in the present case. A new physical phenomenon
specific to the present black-hole is found : the quantum renormalization of
the speed of light. We find $c_{quantum} = \sqrt{{k\o{k-2}}}~c_{classical}$,
where $k$ is the integer in front of the WZW action. This feature is, however,
a pathology. Only for $ k \to \infty$ the pathology disappears (although the
conformal anomaly is present). We analyze all the classical euclidean string
solutions and exactly compute the quantum partition function. No critical
Hagedorn temperature appears here.
