We report measurements of the acoustic and optical phason dispersion curves associated with the lattice of partial dislocations on the reconstructed (111) surface of gold. Our measurements of these low energy (<0.5 meV) weakly dispersive modes have been enabled by the very high resolution of the novel helium spin-echo technique. The results presented here constitute the first measurement of the phason dispersion of misfit dislocations, and possibly of excitations associated with any type of crystalline dislocations.