We sum up the next-to-leading logarithmic corrections to the heavy-quarkonium
hyperfine splitting using the nonrelativistic renormalization group.
On the basis of this result, we predict the mass of the $\eta_b$ meson to be
$M(\eta_b)=9421 \pm 10\,{(\rm th)} \,{}^{+9}_{-8}\,(\delta\alpha_s)~{\rm MeV}$.
The experimental measurement of $M(\eta_b)$ with a few MeV error would be
sufficient to determine $\alpha_s(M_Z)$ with an accuracy of $\pm 0.003$.
The use of the nonrelativistic renormalization group is mandatory to reproduce
the experimental value of the hyperfine splitting in charmonium.