The B-10(p,alpha(0))Be-7 bare nucleus astrophysical S(E) factor has been measured for the first time at energies from about 100 keV down to about 5 keV by means of the Trojan Horse method (THM). In this energy region, the S(E) factor is strongly dominated by the 8.699 MeV C-11 level (J(pi) = 5/2(+)), producing an s-wave resonance centered at about 10 keV in the entrance channel. Up to now, only the high-energy tail of this resonance has been measured, while the low-energy trend is extrapolated from the available direct data. The THM has been applied to the quasifree H-2(B-10,alpha(0) Be-7)n reaction induced at a boron-beam energy of 24.5 MeV. An accurate analysis leads to the determination of the B-10(p,alpha(0))Be-7 S(E) factor and of the corresponding electron screening potential U-e, thus giving for the first time an independent evaluation of it.