We consider a one-dimensional directed polymer in a random potential which is characterized by Gaussian statistics with finite radius local correlations. It is shown that the well known Kardar's solution obtained originally for a directed polymer with -correlated random potential can be applied for the description of the present system only in the high-temperature limit. For the low-temperature limit we have obtained a new solution which is described by the one-step replica symmetry breaking. For the mean square deviation of the directed polymer of linear size L it provides the usual scaling with the wandering exponent and the temperature-independent prefactor.