Discovery of the cosmic reionization epoch would represent a significant milestone in cosmology. We present Keck spectroscopy of the quasar SDSS 1044-0125, at z = 5.73. The spectrum shows a dramatic increase in the optical depth at observed wavelengths lambda >~7550 A, corresponding to z_abs >~ 5.2. Only a few small, narrow transmission regions are present in the spectrum beyond that point, and out to the redshifts where the quasar signal begins. We interpret this result as a signature of the trailing edge of the cosmic reionization epoch, which we estimate to occur around <z> ~ 6 (as indeed confirmed by subsequent observations by Becker et al.), and extending down to z ~ 5.2. This behavior is expected in the modern theoretical models of the reionization era, which predict a patchy and gradual onset of reionization. The remaining transmission windows we see may correspond to the individual reionization bubbles (Stromgren spheres) embedded in a still largely neutral intergalactic medium, intersected by the line of sight to the quasar. Future spectroscopic observations of quasars at comparable or larger redshifts will provide a more detailed insight into the structure and extent of the reionization era.