  We present a new formulation of one of the major radiative corrections to the
weak charge of the proton -- that arising from the axial-vector hadron part of
the $\gamma Z$ box diagram, $\Re{\rm e}\, \Box_{\gamma Z}^{\rm A}$. This
formulation, based on dispersion relations, relates the $\gamma Z$
contributions to moments of the $F_3^{\gamma Z}$ interference structure
function. It has a clear connection to the pioneering work of Marciano and
Sirlin, and enables a systematic approach to improved numerical precision.
Using currently available data, the total correction from all intermediate
states is $\Re{\rm e}\, \Box_{\gamma Z}^{\rm A} = 0.0044(4)$ at zero energy,
which shifts the theoretical estimate of the proton weak charge from
$0.0713(8)$ to $0.0705(8)$. The energy dependence of this result, which is
vital for interpreting the Q$_{\rm weak}$ experiment, is also determined.
