  We present the first and complete dispersion relation analysis of the inner
radiative corrections to the axial coupling constant $g_A$ in the neutron
$\beta$-decay. Using experimental inputs from the elastic form factors and the
spin-dependent structure function $g_1$, we determine the contribution from the
$\gamma W$-box diagram to a precision better than $10^{-4}$. Our calculation
indicates that the inner radiative corrections to the Fermi and the
Gamow-Teller matrix element in the neutron $\beta$-decay are almost identical,
i.e. the ratio $\lambda=g_A/g_V$ is almost unrenormalized. With this result, we
predict the bare axial coupling constant to be
{$\mathring{g}_A=-1.2754(13)_\mathrm{exp}(2)_\mathrm{RC}$} based on the PDG
average $\lambda=-1.2756(13)$
