Gamma-ray bursts (GRBs) and their afterglows are the most brilliant transient events in the Universe. Both the bursts themselves and their afterglows have been predicted to be visible out to redshifts of z approximately 20, and therefore to be powerful probes of the early Universe. The burst GRB 000131, at z = 4.50, was hitherto the most distant such event identified. Here we report the discovery of the bright near-infrared afterglow of GRB 050904 (ref. 4). From our measurements of the near-infrared afterglow, and our failure to detect the optical afterglow, we determine the photometric redshift of the burst to be z = 6.39 - 0.12 + 0.11 (refs 5-7). Subsequently, it was measured spectroscopically to be z = 6.29 +/- 0.01, in agreement with our photometric estimate. These results demonstrate that GRBs can be used to trace the star formation, metallicity, and reionization histories of the early Universe.