Measurement of the branching fractions of the radiative leptonic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi></mml:math>decays<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>e</mml:mi><mml:mi>γ</mml:mi><mml:mi>ν</mml:mi><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo accent="true" stretchy="false">¯</mml:mo></mml:mover></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>μ</mml:mi><mml:mi>γ</mml:mi><mml:mi>ν</mml:mi><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo accent="true" stretchy="false">¯</mml:mo></mml:mover></mml:math>at<i>BABAR</i>

We perform a measurement of the $\ensuremath{\tau}\ensuremath{\rightarrow}l\ensuremath{\gamma}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ ($l=e,\ensuremath{\mu}$) branching fractions for a minimum photon energy of 10 MeV in the $\ensuremath{\tau}$ rest frame, using $431\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of ${e}^{+}{e}^{\ensuremath{-}}$ collisions collected at the center-of-mass energy of the $\mathrm{\ensuremath{\Upsilon}}(4S)$ resonance with the BABAR detector at the PEP-II storage rings. We find $\mathcal{B}(\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\mu}\ensuremath{\gamma}\ensuremath{\nu}\overline{\ensuremath{\nu}})=(3.69\ifmmode\pm\else\textpm\fi{}0.03\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}3}$ and $\mathcal{B}(\ensuremath{\tau}\ensuremath{\rightarrow}e\ensuremath{\gamma}\ensuremath{\nu}\overline{\ensuremath{\nu}})=(1.847\ifmmode\pm\else\textpm\fi{}0.015\ifmmode\pm\else\textpm\fi{}0.052)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}2}$, where the first quoted error is statistical and the second is systematic. These results are substantially more precise than previous measurements.