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 $M(B^*_c)-M(B_c)=65 \pm 24\,{(\rm
th)}\,{}^{+19}_{-16}\,(\delta\alpha_s)$~MeV.