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 $sim10^4$ peculiar velocities covering the whole southern hemisphere for galactic latitude $|b| > 10^circ$, 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 $70,h^{-1}{rm Mpc}$. We investigate and correct for potential bias due to the lognormal velocity uncertainties, and verify our method by constructing $Lambda{rm CDM}$ 6dFGSv mock catalogues incorporating the survey selection function. For a hemisphere of radius $50,h^{-1}{rm Mpc}$ we find a bulk flow amplitude of $U=248pm58,{rm km},{rm s}^{-1}$ in the direction $(l,b) = (318^circpm20^circ,40^circpm13^circ)$, and for $70,h^{-1}{rm Mpc}$ we find $U=243pm58,{rm km},{rm s}^{-1}$, in the same direction. Our measurement gives us a constraint on $sigma_8$ 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 $Lambda{rm CDM}$ prediction.