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 $12^3times 32, 16^3times 48$ and $24^3times 64$, 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.
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