We report the discovery of Qatar-6b, a new transiting planet identified by the Qatar Exoplanet Survey (QES). The planet orbits a relatively bright (V. = 11.44), early-K main-sequence star at an orbital period of P similar to 3.506 days. An SED fit to available multi-band photometry, ranging from the near-UV to the mid-IR, yields a distance of d = 101 +/- 6 pc to the system. From a global fit to follow-up photometric and spectroscopic observations, we calculate the mass and radius of the planet to be M-P = 0.67 +/- 0.07 M-J and R-P = 1.06. +/-. 0.07 R-J, respectively. We use multi-color photometric light curves to show that the transit is grazing, making Qatar-6b one of the few exoplanets known in a grazing transit configuration. It adds to the short list of targets that offer the best opportunity to look for additional bodies in the host planetary system through variations in the transit impact factor and duration.