  We report an improved low-energy extrapolation of the cross section for the
process Beryllium-7+proton -> Boron-8+photon, which determines the Boron-8
neutrino flux from the Sun. Our extrapolant is derived from Halo Effective
Field Theory (EFT) at next-to-leading order. We apply Bayesian methods to
determine the EFT parameters and the low-energy S-factor, using measured cross
sections and scattering lengths as inputs. Asymptotic normalization
coefficients of Boron-8 are tightly constrained by existing radiative capture
data, and contributions to the cross section beyond external direct capture are
detected in the data at E < 0.5 MeV. Most importantly, the S-factor at zero
energy is constrained to be S(0)= 21.3 + - 0.7 eV b, which is an uncertainty
smaller by a factor of two than previously recommended. That recommendation was
based on the full range for S(0) obtained among a discrete set of models judged
to be reasonable. In contrast, Halo EFT subsumes all models into a controlled
low-energy approximant, where they are characterized by nine parameters at
next-to-leading order. These are fit to data, and marginalized over via Monte
Carlo integration to produce the improved prediction for S(E).
