We present a lattice NRQCD study of the B meson decay constant in the quenched approximation with emphasis given to the scaling behavior. The NRQCD action and the heavy-light axial vector current we use include all terms of order (Formula presented) and the perturbative (Formula presented) and (Formula presented) corrections. Using simulations at three values of couplings (Formula presented) 5.9, and 6.1 on lattices of sizes (Formula presented) (Formula presented) and (Formula presented) we find a significant a dependence disappears in (Formula presented) if the (Formula presented) correction is included in the axial vector current. We observe that (Formula presented) is the window where systematic errors are expected to be minimum within one-loop improved theory. Our final results are (Formula presented) MeV, (Formula presented) MeV, and (Formula presented) where the first error is statistical, the second is systematic, and the third is due to the uncertainty of the strange quark mass, while quenching errors are not included. © 2000 The American Physical Society.