  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
$\Lambda^2=6.92\times(1\pm0.002^{+0.014}_{-0.009})$ where the first uncertainty
is due to experimental errors and the second uncertainty is due to theoretical
uncertainties. Our value of $\Lambda^2$ 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.
