  Zero temperature limit in (1+1) directed polymers with finite range
correlated random potential is studied. In terms of the standard replica
technique it is demonstrated that in this limit the considered system reveals
the one-step replica symmetry breaking structure similar to the one which takes
place in the Random Energy Model. In particular, it is shown that at the
temperature $T_{*} \sim (u R)^{1/3}$ (where $u$ and $R$ are the strength and
the correlation length of the random potential) there is a crossover from the
high- to the low-temperature regime. Namely, in the high-temperature regime at
$T >> T_{*}$ the model is equivalent to the one with the $\delta$-correlated
potential where the non-universal prefactor of the free energy is proportional
to $T^{-2/3}$, while at $T << T_{*}$ this non-universal prefactor saturates at
a finite (temperature independent) value.
