The superionic conducting phase of alpha-AgI, which is thermodynamically stable only above 147 degrees C, was frozen at room temperature in AgI-Ag2O-B2O3 glasses prepared by twin-roller rapid quenching. Such alpha-AgI frozen samples exhibited extremely high ion conductivities of about 10(-2)-10(-1) S cm(-1) at room temperature. Spherical particles of alpha-AgI (about 30 nm in diameter) were observed to be homogeneously dispersed in a glass matrix. Heating the composite up to temperatures higher than 80 degrees C, which is near the glass transition temperature of the matrix glass, brought about the relaxation of the lattice strain of alpha-AgI originally generated by the presence of the rigid glass matrix, and accelerated the alpha- to beta-phase transformation. At low temperatures, much larger lattice strain, instead of the alpha-->beta transformation, was observed below -10 degrees C, where the activation energy for conduction changed abruptly and the heat capacity was maximized. Such anomalies at around -10 degrees C were associated with the ordering of Ag+ ions in the frozen alpha-AgI.