  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.
