The fusion reactions10B(p, α)7Be and11B(p, α)8Be have been studied over the c.m. energy range E=17 to 134 keV using intense proton beams and thick solid targets. In the case of11B(p, α)8Be the low-energy data in terms of the astrophysical S(E) factor show an exponential enhancement (up to a factor of 1.9) due to the effects of electron screening, where the deduced screening potential is larger than expected. In the case of10B(p, α)7Be the low-energy data exhibit an enhancement by more than a factor of 200, which cannot be explained by the effects of electron screening. The enhancement arises here from the high-energy tail of an expecteds-wave resonance at ER=10 keV. The results offer an improved prospect for this reaction as advanced fuel in future fusion reactors than previously envisioned.