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 γZ box diagram, ℜe□(γZ)(A). This formulation, based on dispersion relations, relates the γZ contributions to moments of the F(3)(γZ) 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□(γZ)(A)=0.004 4(4) at zero energy, which shifts the theoretical estimate of the proton weak charge from 0.071 3(8) to 0.070 5(8). The energy dependence of this result, which is vital for interpreting the Q(weak) experiment, is also determined.