  Within the context of a super-critical (Liouville) string, we discuss
(target-space) two-dimensional string cosmology. A numerical analysis indicates
that the identification of time with the Liouville mode results in an expanding
universe with matter which exhibits an inflationary phase, and `graceful exit'
from it, tending asymptotically to a flat-metric fixed point.This fixed point
is characterized by a dilaton configuration which, depending on the initial
conditions, either decreases linearly with the cosmic time, or is a finite
constant. This implies that, in contrast to the critical string case, the
string coupling remains bounded during the exit from the inflationary phase,
and, thus, the pertinent dynamics can be reliably described in terms of a
tree-level string effective action. The r\^ole of matter in inducing such
phenomena is emphasized. It is also interesting to note that the asymptotic
value of the vacuum energy, which in the $\sigma$-model framework is identified
with the `running' central charge deficit, depends crucially on the set of
initial conditions. Thus, although preliminary, this toy model seems to share
all the features expected to characterize a phenomenologically acceptable
cosmological string model.
