We present high signal-to-noise ratio Gemini and MMT spectroscopy of the optical afterglow of the gamma-ray burst (GRB) 130606A at redshift z=5.913, discovered by Swift. This is the first high-redshift GRB afterglow to have spectra of comparable quality to those of z~6 quasars. The data exhibit a smooth continuum at near-infrared wavelengths that is sharply cut off blueward of 8410 Angs due to absorption from Ly-alpha at redshift z~5.91, with some flux transmitted through the Ly-alpha forest between 7000-7800 Angs. We use column densities inferred from metal absorption lines to constrain the metallicity of the host galaxy between a lower limit of [Si/H]>-1.7 and an upper limit of [S/H]<-0.5 set by the non-detection of S II absorption. We demonstrate consistency between the dramatic evolution in the transmission fraction of Ly-alpha seen in this spectrum over the redshift range z=4.9 to 5.85 with that previously measured from observations of high-redshift quasars. There is an extended redshift interval of Delta-z=0.12 in the Ly-alpha forest at z=5.77 with no detected transmission, leading to a 3-sigma upper limit on the mean Ly-alpha transmission fraction of <0.2% (or tau_eff(Ly-alpha) > 6.4). This is comparable to the lowest-redshift Gunn-Peterson troughs found in quasar spectra. We set a 2-sigma upper limit of 0.11 on the neutral fraction of the IGM at the redshift of the GRB from the lack of a Ly-alpha red damping wing, assuming a model with a constant neutral density. Some Ly-beta and Ly-gamma transmission is detected in this redshift window, indicating that it is not completely opaque, and hence that the IGM is nonetheless mostly ionized at these redshifts. GRB 130606A thus for the first time realizes the promise of GRBs as probes of the first galaxies and cosmic reionization.
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