  We present the results of a search for $B^{+}\to\gamma \ell^{+}\nu_{\ell}$,
where $\ell = e, \mu$. We use a sample of 232 million $B\bar{B}$ pairs recorded
at the $\Upsilon(4S)$ with the BaBar detector at the PEP-II $B$ Factory. We
measure a partial branching fraction $\Delta\mathcal{B}$ in a restricted region
of phase space that reduces the effect of theoretical uncertainties, requiring
the lepton energy to be between 1.875 and 2.850GeV, the photon energy to be
between 0.45 and 2.35GeV, and the cosine of the angle between the lepton and
photon momenta to be less than -0.36, with all quantities computed in the
$\Upsilon(4S)$ center-of-mass frame. We find $\Delta\mathcal{B}(B^{+}\to\gamma
\ell^{+}\nu_{\ell}) = (-0.3^{+1.3}_{-1.5} ({stat}) \pm 0.6 ({syst}) \pm 0.1
({th})) \times 10^{-6}$, assuming lepton universality. Interpreted as a 90%
C.L. Bayesian upper limit, the result corresponds to $1.7\times10^{-6}$ for a
prior flat in amplitude, and $2.3\times10^{-6}$ for a prior flat in branching
fraction.
