  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%.
