We re-evaluate the matrix element for the proton-proton reaction which is important for stellar-evolution calculations and for the solar-neutrino problem. We self-consistently determine the effect of vacuum polarization on the matrix element by first correcting the low-energy scattering data to account for vacuum polarization. We then calculate the proton-proton wave function by integrating the Schrodinger equation with vacuum polarization included. We use improved data for proton-proton scattering and for the deuteron wave function. We evaluate the uncertainties that are due to experimental error and estimate those that are due to theoretical inadequacies. We estimate the theoretical uncertainty by using six different deuteron potentials and five different proton-proton potentials. Vacuum polarization decreases the calculated value by $0.6_{-0.4}^{+0.1}$\%. The complete result is $��^2=6.92\times(1\pm0.002^{+0.014}_{-0.009})$ where the first uncertainty is due to experimental errors and the second uncertainty is due to theoretical uncertainties. Our value of $��^2$ is 2\% smaller than the value obtained in 1969 by Bahcall and May. The improved calculations of the rate of the $pp$ reaction described here increase slightly the predicted event rates for the chlorine and the Kamiokande solar-neutrino experiments.

22 pages, (phyzzx.tex, 4 figures not included, available upon request), IASSNS-HEP-93/21, IASSNS-AST-93/20