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 +/- 0.01 M_Jup and radius of 1.22 ^{+0.11}_{-0.08} R_Jup. There is a linear trend in the radial velocities of 55+/-4 m/s/y indicating the presence of a long-period third body in the system with a mass > 0.45 M_Jup at a distance of >1.2 AU 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%.
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