  The masses of spin-singlet and spin-triplet S-wave mesons containing a single
heavy quark are computed in the quenched approximation. The light quark action
and gauge field action are both classically-improved and tadpole-improved, and
the couplings to the heavy quark are organized by the 1/M expansion of
tadpole-improved NRQCD. At each of two lattice spacings, near 0.22fm and
0.26fm, meson masses are obtained for heavy quarks spanning the region between
charmed and bottom mesons. Results up to O(1/M), O(1/M^2) and O(1/M^3) are
displayed separately, so that the convergence of the heavy quark expansion can
be discussed. Also, the effect of each term in the O(1/M^3) contribution is
computed individually. For bottom mesons the 1/M-expansion appears to be
satisfactory, but the situation for charmed mesons is less clear.
