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 $lambda=-1.2756(13)$
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