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 η b meson to be M ( η b ) = 9421 ± 11 ( th ) + 9 − 8 ( δ α s )     MeV . The experimental measurement of M ( η b ) with a few MeV error would be sufficient to determine α s ( M Z ) with an accuracy of ± 0.003 . For the hyperfine splitting in charmonium, the use of the nonrelativistic renormalization group brings the perturbative prediction significantly closer to the experimental figure.