We determine the small-$x$ asymptotics of the gluon
helicity distribution in a proton at leading order in
perturbative QCD at large $N_c$. To achieve this, we begin by
evaluating the dipole gluon helicity TMD at small $x$. We then
construct and solve novel small-$x$ large-$N_c$ evolution
equations for the operator related to the dipole gluon
helicity TMD. Our main result is the small-$x$ asymptotics
for the gluon helicity distribution:
\begin{align}\label{dG_final}
\Delta G \sim \left(
\frac{1}{x} \right)^{\alpha_h^G} \ \ \ \mbox{with} \ \ \ \alpha_h^G =
\frac{13}{4 \sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}}
\approx 1.88 \, \sqrt{\frac{\as \, N_c}{2 \pi}}.
\end{align}
We note that the power $\alpha_h^G$ is approximately 20$\%$ lower than
the corresponding power $\alpha_h^q$ for the small-$x$ asymptotics of
the quark helicity distribution defined by
\begin{align}\label{dq_final}
\Delta q \sim \left(
\frac{1}{x} \right)^{\alpha_h^q} \ \ \ \mbox{with} \ \ \ \alpha_h^q =
\frac{4}{\sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}} \approx 2.31 \,
\sqrt{\frac{\as \, N_c}{2 \pi}}
\end{align}
found in our earlier work.