  We study the graceful exit problem and the role of the stress-energy-momentum
tensors in the two-dimensional string cosmology. The one-loop quantum
correction of conformal fields is incorporated in the arbitrary large $N$ limit
to ensure exact quantum solvability. The only solution which gives the bounded
curvature with the asymptotic flatness is restricted to the first branch under
some conditions. However, even in this case, the accelerating expansion is
forever. We show that the only nonvanishing quantum stress-momentum tensor is
the pressure part($T_{xx}$) which is of relevance to the dynamical evolution of
the universe in the comoving coordinate. The quantum energy part is zero since
the negative contribution of the induced conformal matters always cancels the
positive quantity of the induced dilaton part in terms of the constraint
equation.
