We have experimentally studied the time-evolution of the exciton population in a higher subband of GaAs quantum wells, below the free carrier continuum. The lifetime of the exciton formed by an electron of the lowest subband and a heavy hole of the second subband in GaAs quantum wells is determined by time-resolved luminescence at 130 +/- 20 ps. This result is consistent with theoretical estimations of intersubband scattering by acoustic phonon emission. The exciton lifetime in the second heavy-hole subband is considerably longer than reported values of the recombination time in the lowest exciton state at k equals 0. The excitons in the higher subband at k equals 0 can be excited selectively without exciting the lower subband at k > 0. From these findings we conclude that subband transitions of excitons in quantum wells represent a new appealing concept for optically pumped coherent sources in the meV range.