Inelastic-neutron-scattering measurements have been performed on the high-temperature cubic perovskite phase of LaAlO3. Dispersion curves for phonons with q along  establish that near the zone corner (R point), the frequency of the lowest branch is both anomalously low and temperature-dependent. The nature of the associated scattering changes continuously from inelastic to quasi-elastic as the temperature is lowered, indicating that the soft-phonon modes become overdamped. The intensity of this quasi-elastic scattering exhibits a Curie-law divergence as the temperature is lowered further toward the transformation temperature of (535±5)°C. The displacements associated with the rhombohedral low-temperature structure result from the condensation of a particular linear combination of these soft triply degenerate Γ25 modes, as proposed by Cochran and Zia. © 1969 The American Physical Society.