  We consider the directed polymer model in the weak disorder phase under the
assumption that the partition function is $L^p$-bounded for some $p>1+2/d$. We
prove a local limit theorem for the polymer measure, i.e., that the
point-to-point partition function can be approximated by two point-to-plane
partition functions at the start- and endpoint. We furthermore show that for
environments with finite support the required $L^p$-boundedness holds in the
whole weak disorder phase, except possible for the critical value $\beta_{cr}$.
Some consequences of the local limit theorem are also discussed.
