Heat capacities of (AgI)(x)(AgPO3)(1-x)(x = 0, 0.500, 0.600 and 0.650) were precisely measured between 13 and 400 K with an adiabatic calorimeter. beta-Glass transitions due to the freezing-in of rearrangement of Ag+ ions were observed for the glasses with x = 0.500, 0.600 and 0.650 at 85 +/- 1, 76 +/- 1 and 74 +/- 1 K, respectively. The energy differences between the two lowest-in-energy accessible sites to each Ag+ ion were estimated to be 6.1 +/- 0.6, 5.3 +/- 0.3 and 5.2 +/- 0.3 kJ mol(-1) for x = 0.500, 0.600 and 0.650 glasses, respectively. The conductometry and the dielectric measurements were found to see the same mode of Ag+ motion as the calorimetry. The activation energies for the diffusional motions were evaluated to be 29.3, 26.2 and 25.2 kJ mol(-1) for x = 0.500, 0.600 and 0.650 glasses, respectively. It was deduced that the positional order/disorder process of Ag+ ions is essential to the fast ionic conduction in the glasses and that the AgI aggregate region naturally formed at high AgI compositions mainly contributes to the fast ionic conduction and has some amorphous structure dependent on the development of the region. Finally, the 'mixed electrolyte tissue to amorphous AgI aggregate' model was proposed for the glass structures from the points of view of ionic conduction and positional disorder of Ag+ ions.