The astrophysical S factor for the proton weak capture on He-3 is calculated with correlated-hyperspherical-harmonics bound and continuum wave functions corresponding to realistic Hamiltonians consisting of the Argonne nu (14) or Argonne nu (18) two-nucleon and Urbana-VIII or Urbana-IX three-nucleon interactions. The nuclear weak charge and current operators have vector and axial-vector components that include one- and many-body terms. All possible multipole transitions connecting any of the p-He-3 S- and P-wave channels to the He-4 bound state are considered. The S factor at a p-He-3 center-of-mass energy of 10 keV, close to the Gamow-peak energy, is predicted to be 10.1 x 10(-20) keV b with the AV18/UIX Hamiltonian, a factor of similar or equal to1.5 larger than the value adopted in the standard solar model. The P-wave transitions are found to be important, contributing about 40% of the calculated S factor. The energy dependence is rather weak: the AV18/UIX zero-energy S factor is 9.64 x 10(-20) keV b, only 5% smaller than the 10 keV result quoted above. The model dependence is also found to be weak: the zero-energy S factor is calculated to be 10.2 x 10(-20) keV b with the older AV14/UVIII model, only 6% larger than the AV18/UIX result. Our best estimate for the S factor at 10 keV is therefore (10.1+/-0.6) x 10(-20) keV b, when the theoretical uncertainty due to the model dependence is included. This value for the calculated S factor is not as large as determined in fits to the Super-Kamiokande data in which the hrp flux normalization is free. However, the precise calculation of the S factor and the consequent absolute prediction for the hrp neutrino flux will allow much greater discrimination among proposed solar neutrino oscillation solutions.