We have used high resolution stimulated Brillouin gain spectroscopy to monitor the nonequilibrium relaxation dynamics of supercooled glycerol near its glass transition temperature. After a temperature quench from 192.4 to 179.8 K, the Brillouin frequency shift increased over a period of 17 d from 16.877+/-0.005 GHz to 17.195+/-0.005 GHz, the equilibrium value expected for the final temperature. The relaxation of the shift appears to proceed in three distinct stages. The first and most rapid stage tracks the time dependence of the temperature. This is followed by a slower relaxation process which is well characterized bq a stretched exponential function exp[-(t/tau)(beta)]. The values determined for tau (12+/-2 ks) and beta (0.61+/-0.09) are consistent with values predicted from recent dielectric and enthalpy relaxation studies of glycerol in the same temperature range, as long as allowances are made for the nonlinear nature of the nonequilibrium experiment. The third and final stage in the relaxation is approximately exponential with a time constant of 430+/-40 ks. We speculate that this ultraslow process may have its origin in the relaxation of mechanical strain built up in the sample during the temperature quench, or that it could result from the nonlinear nature of the experiment and the heterogeneous nature of dynamics in supercooled liquids. A comparison with other methods (dilatometry, probe fluorescence) for studying nonequilibrium relaxation in supercooled liquids shows our technique to be comparable in sensitivity. Finally, the Brillouin linewidth showed no change after an initial rapid decrease to 18+/-2 MHz (half width at half maximum), indicating a lack of significant spatial heterogeneity in the physical properties and/or relaxation dynamics on length scales of the order of the acoustic wavelength (0.2 mu m) or longer.