  Lattice NRQCD with leading finite lattice spacing errors removed is used to
calculate decay constants of mesons made up of heavy quarks. Quenched
simulations are done with a tadpole improved gauge action containing plaquette
and six-link rectangular terms. The tadpole factor is estimated using the
Landau link. For each of the three values of the coupling constant considered,
quarkonia are calculated for five masses spanning the range from charmonium
through bottomonium, and one set of quark masses is tuned to the B(c).
"Perturbative" and nonperturbative meson masses are compared. One-loop
perturbative matching of lattice NRQCD with continuum QCD for the heavy-heavy
vector and axial vector currents is performed. The data are consistent with the
vector meson decay constants of quarkonia being proportional to the square root
of their mass and the B(c) decay constant being equal to 420(13) MeV.
