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We present high signal-to-noise ratio Gemini and MMT spectroscopy of theoptical 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 havespectra of comparable quality to those of z~6 quasars. The data exhibit asmooth continuum at near-infrared wavelengths that is sharply cut off bluewardof 8410 Angs due to absorption from Ly-alpha at redshift z~5.91, with some fluxtransmitted through the Ly-alpha forest between 7000-7800 Angs. We use columndensities inferred from metal absorption lines to constrain the metallicity ofthe 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 demonstrateconsistency between the dramatic evolution in the transmission fraction ofLy-alpha seen in this spectrum over the redshift range z=4.9 to 5.85 with thatpreviously measured from observations of high-redshift quasars. There is anextended redshift interval of Delta-z=0.12 in the Ly-alpha forest at z=5.77with no detected transmission, leading to a 3-sigma upper limit on the meanLy-alpha transmission fraction of <0.2% (or tau_eff(Ly-alpha) > 6.4). This iscomparable to the lowest-redshift Gunn-Peterson troughs found in quasarspectra. We set a 2-sigma upper limit of 0.11 on the neutral fraction of theIGM 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-gammatransmission is detected in this redshift window, indicating that it is notcompletely opaque, and hence that the IGM is nonetheless mostly ionized atthese redshifts. GRB 130606A thus for the first time realizes the promise ofGRBs as probes of the first galaxies and cosmic reionization.

GRB 130606A as a Probe of the Intergalactic Medium and the Interstellar Medium in a Star-forming Galaxy in the First Gyr After the Big Bang