  By using the Highly Improved Staggered Quark formalism to handle charm,
strange and light valence quarks in full lattice QCD, and NRQCD to handle
bottom valence quarks we are able to determine accurately ratios of the B meson
vector-pseudoscalar mass splittings, in particular,
(m(B*_c)-m(B_c))/(m(B*_s)-m(B_s)). We find this ratio to be 1.15(15), showing
the `light' quark mass dependence of this splitting to be very small. Hence we
predict m(B_c*) = 6.330(7)(2)(6) GeV where the first two errors are from the
lattice calculation and the third from existing experiment. This is the most
accurate prediction of a gold-plated hadron mass from lattice QCD to date.
