  B->gamma e nu transitions have recently been studied in the framework of QCD
factorization. The attractiveness of this channel for such an analysis lies in
the fact that, at least in the heavy quark limit, the only hadron involved is
the B meson itself, so one expects a very simple description of the form factor
in terms of a convolution of the B meson distribution amplitude with a
perturbative kernel. This description, however, does not include contributions
suppressed by powers of the b quark mass. In this letter, we calculate
corrections to the factorized expression which are induced by the ``soft''
hadronic component of the photon. We demonstrate that the power-suppression of
these terms is numerically not effective for physical values of the $b$ quark
mass and that they increase the form factor by about 30% at zero momentum
transfer. We also derive a sum rule for lambda_B, the first negative moment of
the B meson distribution amplitude, and find lambda_B = 0.6 GeV (to leading
order in QCD).
