We construct an exact analytic solution of the revised small-$x$ helicity evolution equations derived previously. The equations we solve are obtained in the large-$N_c$ limit (with $N_c$ the number of quark colors) and are double-logarithmic (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling constant and $x$ the Bjorken $x$ variable). Our solution provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function, with their leading small-$x$ asymptotics given by
\begin{align}
\Delta \Sigma (x, Q^2) \sim \Delta G (x, Q^2)
\sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{\alpha_h} , \notag
\end{align}
where the exact analytic expression we obtain for the intercept $\alpha_h$ can be approximated by $\alpha_h = 3.66074 \, \sqrt{\frac{\alpha_s \, N_c}{2 \pi}}$. Our solution also yields an all-order (in $\alpha_s$) resummed small-$x$ anomalous dimension $\Delta \gamma_{GG} (\omega)$ which agrees with all the existing fixed-order calculations (to three loops). Notably, our anomalous dimension is different from that obtained in the infrared evolution equation framework developed earlier by Bartels, Ermolaev, and Ryskin (BER), with the disagreement starting at four loops. Despite the previously reported agreement at two decimal points based on the numerical solution of the same equations, the intercept of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ given by a different analytic expression from ours with the numerical value of $\alpha_h^{BER} = 3.66394 \, \sqrt{\frac{\alpha_s \, N_c}{2 \pi}}$. We speculate on the origin of this disagreement.