We report on the detection by Swift of GRB 080913, and subsequent optical/near-infrared follow-up observations by GROND which led to the discovery of its optical/NIR afterglow and the recognition of its high-z nature via the detection of a spectral break between the i and z bands. Spectroscopy obtained at the ESO-VLT revealed a continuum extending down to lambda = 9400 A, and zero flux for 7500 A < lambda<9400 A, which we interpret as the onset of a Gunn-Peterson trough at z=6.695+-0.025 (95.5% conf. level), making GRB 080913 the highest redshift GRB to date, and more distant than the highest-redshift QSO. We note that many redshift indicators which are based on promptly available burst or afterglow properties have failed for GRB 080913. We report on our follow-up campaign and compare the properties of GRB 080913 with bursts at lower redshift. In particular, since the afterglow of this burst is fainter than typical for GRBs, we show that 2m-class telescopes can identify most high-redshift GRBs.