  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
$M(\eta_b)=9419 \pm 11 {(\rm th)} {}^{+9}_{-8} (\delta\alpha_s) MeV$. 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.
