We report the discovery of WASP-34b, a sub-Jupiter-mass exoplanet transiting its 10.4-magnitude solar-type host star (1SWASP J110135.89-235138.4; TYC 6636-540-1) every 4.3177 days in a slightly eccentric orbit (e=0.038+/-0.012). We find a planetary mass of 0.59+/-0.01M_Jup_ and radius of 1.22^+0.11^-0.08_R_Jup. There is a linear trend in the radial velocities of 55+/-4m/s/y indicating the presence of a long-period third body in the system with a mass >0.45M_Jup at a distance of >1.2AU from the host star. This third-body is either a low-mass star, white dwarf, or another planet. The transit depth ((R_P/R_*)^2^=0.0126) and high impact parameter (b=0.90) suggest that this could be the first known transiting exoplanet expected to undergo grazing transits, but with a confidence of only ~80%.