  We compute the form factors parametrizing radiative leptonic decays of heavy
mesons B^+\to\gamma e^+\nu for photon energies much larger than \Lambda_{QCD},
where perturbative QCD methods for exclusive processes can be combined with the
heavy quark effective theory. The form factors can be reliably obtained in this
region in an expansion in powers of \Lambda/E_\gamma. The leading term in this
expansion displays an additional spin symmetry manifested in the equality of
form factors of vector and axial currents. The leading twist form factors can
be written as the convolution of the B meson light-cone wave function with a
hard scattering amplitude, which is explicitly calculated to one-loop order.
The Sudakov double logarithms of the form
(\frac{\alpha_s}{\pi}\log^2\frac{2E_\gamma}{\Lambda})^n are resummed to all
orders. As an application we present a method for determining the CKM matrix
element |V_{ub}| from a comparison of photon spectra in B and D radiative
leptonic decays.
