  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 current we use include all terms of order 1/M
and the perturbative $O(\alpha_s a)$ and $O(\alpha_s/M)$ corrections. Using
simulations at three value of couplings $\beta$=5.7, 5.9 and 6.1 on lattices of
size $12^3\times 32, 16^3\times 48$ and $24^3\times 64$, we find no significant
$a$ dependence in $f_B$ if the $O(\alpha_s a)$ correction is included in the
axial current. We obtain $f_B = 167(7)(15)$ MeV, $f_{B_s}=
191(4)(17)(^{+4}_{-0})$ MeV and $f_{B_s}/f_B =1.15(3)(1)(^{+3}_{-0})$, with the
first error being statistical, the second systematic, and the third due to
uncertainty of strange quark mass, while quenching errors being not included.
