Previous experiments have shown that the elastic moduli of pure fcc metals are lowered by the presence of mobile dislocation loops. The amount of this lowering is often called the modulus defect. Results of accurate modulus measurements from 20\ifmmode^\circ\else\textdegree\fi{}K down to 0.1\ifmmode^\circ\else\textdegree\fi{}K using liquid-helium and adiabatic-demagnetization techniques show that this modulus defect is temperature-dependent and decreases as the temperature approaches absolute zero. Thus the mobility of dislocations must be decreasing at these low temperatures with most of the decrease confined to temperatures below 1\ifmmode^\circ\else\textdegree\fi{}K. In fact, the decrease is most rapid at 0.1\ifmmode^\circ\else\textdegree\fi{}K. No attenuation changes were observed which could be directly associated with the modulus changes, thereby ruling out the possibility that the decreased mobility of dislocations arises from a thermally activated process with a unique but small activation energy. The present modulus measurements are in apparent disagreement with the often quoted statement that the elastic moduli must approach absolute zero with zero slope. However, the absolute zero of temperature has not been obtained, and there is no reason to expect that the observed dislocation effects should not ultimately approach zero with a zero slope at temperatures much below 0.1\ifmmode^\circ\else\textdegree\fi{}K.