  The cross section for the proton weak capture reaction $^1H(p,e^+\nu_e)^2H$
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 $\beta$ 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 $\Lambda^2(E=0)$ 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.
