  We compute the hyperfine splitting in a heavy quarkonium composed of
different flavors in next-to-leading logarithmic approximation using the
nonrelativistic renormalization group. We predict the mass difference of the
vector and pseudoscalar charm-bottom mesons to be $M(B^*_c)-M(B_c)=46 \pm 15
{(\rm th)} {}^{+13}_{-11} (\delta\alpha_s)$ MeV.
