We measure the bulk flow of the local Universe using the 6dF Galaxy Survey peculiar velocity sample (6dFGSv), the largest and most homogeneous peculiar velocity sample to date. 6dFGSv is a Fundamental Plane sample of ~104 peculiar velocities covering the whole Southern hemisphere for galactic latitude |b| > 10°, out to redshift z = 0.0537.We apply the 'minimum variance' bulk flow weighting method, which allows us to make a robust measurement of the bulk flow on scales of 50 and 70 h-1 Mpc. We investigate and correct for potential bias due to the lognormal velocity uncertainties, and verify our method by constructing Λ cold dark matter (ΛCDM) 6dFGSv mock catalogues incorporating the survey selection function. For a hemisphere of radius 50 h-1 Mpc we find a bulk flow amplitude of U = 248 ± 58 km s-1 in the direction (l, b) = (318° ± 20°, 40° ± 13°), and for 70 h-1 Mpc we find U = 243 ± 58 km s-1, in the same direction. Our measurement gives us a constraint on σ8 of 1.01+1.07-0.58. Our results are in agreement with other recent measurements of the direction of the bulk flow, and our measured amplitude is consistent with a ΛCDM prediction.