The diffusion mechanism for Cu+ was studied by means of NMR, powder X-ray diffraction and AC conductivity measurements. The satellite spectra of the Cu-63- and Al-27-NMR varied anomalously with temperature. The quadrupole coupling constant of the Cu-63 nucleus and the asymmetric parameter of the Al-27 nucleus changed remarkably with decreasing temperature and the quadrupole coupling constant of the Al-27 nucleus did not change so much. From calculation of the electric-field-gradient tensors, it was confirmed that these temperature dependencies are produced by both Cui diffusion and variation of the lattice constants. The diffusion path of Cu+ in CuAlBr4 was confirmed by NMR experiments: Cu+ jumps between normal sites via an interstitial site. The ionic conductivity was evaluated by assuming that the diffusion mechanism and its value were consistent with the observed AC conductivity.