We give results for the B and the D meson spectrum using NRQCD on the latti
ce in the quenched approximation. The masses of radially and orbitally exci
ted states are calculated as well as S-wave hyperfine and P-wave fine struc
ture. Radially excited P states are observed for the first time. Radial and
orbital excitation energies match well to experiment, as does the strange-
nonstrange S-wave splitting. We compare the light and heavy quark mass depe
ndence of various splittings to experiment. Our B results cover a range in
lattice spacings of more than a factor of two. Our D results are from a sin
gle lattice spacing and we compare them to numbers in the literature from f
iner lattices using other methods. We see no significant dependence of phys
ical results on the lattice spacing.