  In this Letter, we analytically solve the evolution equations for the
small-$x$ asymptotic behavior of the (flavor singlet) quark helicity
distribution in the large-$N_c$ limit. These evolution equations form a set of
coupled integro-differential equations, which previously could only be solved
numerically. This approximate numerical solution, however, revealed simplifying
properties of the small-$x$ asymptotics, which we exploit here to obtain an
analytic solution. We find that the small-$x$ power-law tail of the quark
helicity distribution scales as $\Delta q^S (x, Q^2) \sim \left(\tfrac{1}{x}
\right)^{\alpha_h}$ with $\alpha_h = \tfrac{4}{\sqrt{3}} \sqrt{\tfrac{\alpha_s
N_c}{2\pi}}$, in excellent agreement with the numerical estimate $\alpha_h
\approx 2.31\sqrt{\tfrac{\alpha_s N_c}{2\pi}}$ obtained previously. We then
verify this solution by cross-checking the predicted scaling behavior of the
auxiliary "neighbor dipole amplitude" against the numerics, again finding
excellent agreement.
