The cross section for the proton weak capture reaction $^1H(p,e^+��_e)^2H$ is calculated with wave functions obtained from a number of modern, realistic high-precision interactions. To minimize the uncertainty in the axial two-body current operator, its matrix element has been adjusted to reproduce the measured Gamow-Teller matrix element of tritium $��$ decay in model calculations using trinucleon wave functions from these interactions. A thorough analysis of the ambiguities that this procedure introduces in evaluating the two-body current contribution to the pp capture is given. Its inherent model dependence is in fact found to be very weak. The overlap integral $��^2(E=0)$ for the pp capture is predicted to be in the range 7.05--7.06, including the axial two-body current contribution, for all interactions considered.

17 pages RevTeX (twocolumn), 5 postscript figures