Motivated by the first search for the rare charged-current B decay to four leptons, ell bar{nu }_ell ell ^{(prime )} bar{ell }^{(prime )}, we calculate the decay amplitude with factorization methods. We obtain the Brightarrow gamma ^* form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in Lambda _mathrm{QCD}/m_b to next-to-leading order in alpha _s, and at mathcal {O}(alpha _s^0) at next-to-leading power. Our calculations predict branching fractions of a few times 10^{-8} in the ell ^{(prime )} bar{ell }^{(prime )} mass-squared bin up to q^2=1~GeV^2 with n_+q>3~GeV. The branching fraction rapidly drops with increasing q^2. An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment lambda _B of the leading-twist B meson light-cone distribution amplitude. We find that in the small-q^2 bin, the sensitivity to lambda _B is almost comparable to B^- rightarrow mathrm {ell }^- bar{nu }_{mathrm {ell }}gamma when lambda _B is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger q^2.