We perform a measurement of the branching fractions of radiative leptonic tau decays for a minimum photon energy of 10 MeV in the τ rest frame, using 431 inverse femto-barn of electron-positron collisions collected at the center-of-mass energy of the Υ (4S) resonance with the BABAR detector at the PEP-II storage rings. We find BF(τ -> μγν¯ν) = (3.69±0.03±0.10)×10−3, and B(τ -> eγν¯ν) = (1.847±0.015±0.052)×10−2 , where
the first quoted error is statistical, and the second is systematic. These results are substantially more precise than previous measurements.