  We present the result of the search for the rare $B$ meson decay of $B^+ \to
\ell^{\,+} \nu_{\ell} \gamma$ with $\ell =e,\mu$. For the search the full data
set recorded by the Belle experiment of $711 \, \mathrm{fb}^{-1}$ integrated
luminosity near the $\Upsilon (4S)$ resonance is used. Signal candidates are
reconstructed for photon energies $E_{\gamma}$ larger than $1 \, \mathrm{GeV}$
using a novel multivariate tagging algorithm. The novel algorithm fully
reconstructs the second $B$ meson produced in the collision using hadronic
modes and was specifically trained to recognize the signal signature in
combination with hadronic tag-side $B$ meson decays. This approach greatly
enhances the performance. Background processes that can mimic this signature,
mainly charmless semileptonic decays and continuum processes, are suppressed
using multivariate methods. The number of signal candidates is determined by
analyzing the missing mass squared distribution as inferred from the signal
side particles and the kinematic properties of the tag-side $B$ meson. No
significant excess over the background-only hypothesis is observed and upper
limits on the partial branching fraction $ \Delta \mathcal{B} $ with
$E_{\gamma}> 1 \, \mathrm{GeV}$ individually for electron and muon final states
as well as for the average branching fraction of both lepton final states are
reported. We find a Bayesian upper limit of $\Delta \mathcal{B}( B^{+} \to
\ell^{\, +} \nu_{\ell} \gamma) < 3.0 \times 10^{-6}$ at 90% CL and also report
an upper limit on the first inverse moment of the light-cone distribution
amplitude of the $B$ meson of $\lambda_B$ at 90% CL.
