  The $B$-meson light-cone distribution amplitude is an important
non-perturbative quantity arising in the factorization of the amplitudes for
many exclusive decays of $B$ mesons, such as $B^-\to\gamma\,\ell^-\bar\nu$. We
reconsider the renormalization-group (RG) equation satisfied by this function
and present its solution at next-to-leading order (NLO) in RG-improved
perturbation theory in Laplace space and, for the first time, in momentum space
and the so-called diagonal (or dual) space. Since the information needed to
describe the $B$ decay processes at leading order in $\Lambda_{\rm QCD}/m_b$ is
most directly contained in the distribution amplitude in Laplace space
evaluated near the origin, we propose an unbiased parameterization of this
object in terms of a small set of uncorrelated hadronic parameters. Using
recent results on the three-loop anomalous dimension for heavy-light current
operators, we derive an expression for the convolution integral appearing in
the $B^-\to\gamma\,\ell^-\bar\nu$ factorization formula that is explicitly
scale independent, and we evaluate this formula at (approximate) NNLO.
