The mechanical response of niobium and molybdenum during one dimensional shock loading in the weak shock regime is investigated in terms of the Hugoniot elastic limit (dynamic yield) and spall (tensile) strengths. Results indicate that although both metals have high elastic limits of ca. 2 GPa, their responses are very different. Deformation in the weak shock regime in niobium is controlled by both the motion and generation of dislocations, resulting in high spall (dynamic tensile) strengths and ductility. In contrast, molybdenum has low spall strength and ductility, which suggests lower dislocation mobility in this metal. We have also shown that the strain-rate in the rising part of the shock front is related to the stress amplitude by the fourth power, as first shown by Swegle and Grady. Although we have not been able to elucidate further on the power relation, we believe that the scaling factor A is related to a materials ability to accommodate shock imposed plasticity via slip and dislocation generation. Overall, we have used arguments about the Peierls stress in body centred cubic metals to explain these results, with niobium (low Peierls stress) having a high dislocation mobility, resulting in behaviour showing some similarities to face centred cubic metals. Molybdenum, with its much higher Peierls stress has a much lower dislocation mobility, and hence lower spall strengths and ductility.