The diffusion of H and D has been investigated in the Nb0.80Mo0.20 alloy by Gorsky and dipole reorientation/reaction relaxations in the temperature ranges 130-370 K and 70 -100 K, respectively. Between 130 and 370 K the chemical diffusion coefficient at infinite dilution of H and D obeys Arrhenius laws. Differently from the case of pure Nb no deviation from an exponential temperature dependence has been observed below 250 K. The activation energies in the alloy (0.113 +/- 0.003 eV for H and 0.13 +/- 0.02 eV for D) are slightly higher than in pure Nb (0.106 +/- 0.006 eV for H and 0.127 +/- 0.006 for D) for T > 250 K. The H diffusion data were fitted to a relationship provided by a trapping model and the binding energy of H to Mo was found to be as small as 0.009 eV. An internal friction (IF) peak occurring between about 70 and 100 K for frequencies in the range 0.50-6.62 kHz was proved to be due to stress-assisted changes in the directional short-range order of Mo-H(D) bonds. The Einstein diffusion coefficient of H(D) deduced from the IF peak does not match the higher temperature data due to differences in the tunneling regimes of free and trapped H atoms operating over the two temperature ranges. The mismatch is small for D for which the tunneling regime appears to be the same in the free and trapped states. Copyright (C) 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.