We report on follow-up observations of the gamma-ray burst GRB 060927 using the robotic ROTSE-IIIa telescope and a suite of larger aperture ground-based telescopes. An optical afterglow was detected 20 s after the burst, the earliest rest-frame detection of optical emission from any GRB. Spectroscopy performed with the VLT about 13 hr after the trigger shows a continuumbreak at lambda approximate to 8070 angstrom, produced by neutral hydrogen absorption at z approximate to 5.6. We also detect an absorption line at 8158 angstrom, which we interpret as Si II lambda 1260 at z = 5. 467. Hence, GRB 060927 is the second most distant GRB with a spectroscopically measured redshift. The shape of the red wing of the spectral break can be fitted by a damped Ly alpha profile with a column density with log (N-H/cm(-2)) = 22-50 +/- 0.15. We discuss the implications of thiswork for the use ofGRBs as probes of the end of the dark ages and draw threemain conclusions: (1) GRB afterglows originating from z less than or similar to 6 should be relatively easy to detect from the ground, but rapid near-infrared monitoring is necessary to ensure that they are found; (2) the presence of large H I column densities in some GRB host galaxies at z > 5 makes the use of GRBs to probe the reionization epoch via spectroscopy of the red damping wing challenging; and ( 3) GRBs appear crucial to locate typical star-forming galaxies at z > 5, and therefore the type of galaxies responsible for the reionization of the universe.