Glassy materials display numerous important properties which relate to the presence and intensity of the secondary (β) relaxations that dominate the dynamics below the glass transition temperature. However, experimental protocols such as annealing allow little control over the β relaxation for most glasses. Here we report on the β relaxation of toluene in highly stable glasses prepared by physical vapor deposition. At conditions that generate the highest kinetic stability, about 70% of the β relaxation intensity is suppressed, indicating the proximity of this state to the long-sought "ideal glass." While preparing such a state via deposition takes less than an hour, it would require ~3500 years of annealing an ordinary glass to obtain similarly suppressed dynamics.