  We present the differential rates and branching ratios of the radiative
decays $\tau \to l \bar{\nu} \nu \gamma$, with $l=e$ or $\mu$, and $\mu \to e
\bar{\nu} \nu \gamma$ in the Standard Model at next-to-leading order. Radiative
corrections are computed taking into account the full depencence on the mass
$m_l$ of the final charged leptons, which is necessary for the correct
determination of the branching ratios. Only partial agreement is found with
previous calculations performed in the $m_l \to 0$ limit. Our results agree
with the measurements of the branching ratios $\mathcal{B} (\mu \to e \bar{\nu}
\nu \gamma)$ and $\mathcal{B} (\tau \to \mu \bar{\nu} \nu \gamma)$ for a
minimum photon energy of 10 MeV in the $\mu$ and $\tau$ rest frames,
respectively. Babar's recent precise measurement of the branching ratio
$\mathcal{B} (\tau \to e \bar{\nu} \nu \gamma)$, for the same photon energy
threshold, differs from our prediction by 3.5 standard deviations.
