We study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we give a new proof for the central limit theorem (CLT) in probability for the directed polymer model in the interior of the weak disorder phase. We also show that the large deviation rate function agrees with that of the underlying random walk. For the Brownian polymer model, we improve the convergence in the CLT to almost sure convergence in the whole weak disorder phase. The main technical tools are a new moment bound from \cite{J21_1} and a quantitative comparison between the associated martingales at different inverse temperatures.