We measured the 7Be(p,gamma)8B cross section from E(c.m.) = 186 to 1200 keV, with a statistical-plus-systematic precision per point of better than +/-5%. All important systematic errors were measured including 8B backscattering losses. We obtain S17(0) = 22.3+/-0.7(expt)+/-0.5(theor) eV b from our data at E(c.m.)< or =300 keV and the theory of Descouvemont and Baye.