We report on the experimental evidence for a nanosecond timescale spin memory based on nonradiative excitons with large in-plane wave vector. The effect manifests itself in magnetic-field-induced oscillations of the energy of the optically active (radiative) excitons. The oscillations detected by a spectrally resolved pump-probe technique applied to a GaAs/AlGaAs quantum well structure in a transverse magnetic field persist over a timescale, which is orders of magnitude longer than the characteristic decoherence time in the system. The effect is attributed to the spin-dependent electron-electron exchange interaction of the optically active and inactive excitons. The spin relaxation time of the electrons belonging to nonradiative excitons appears to be much longer than the hole spin relaxation time.