  A lattice QCD calculation of the B meson decay constant is presented. In
order to investigate the scaling violation associated with the heavy quark,
parallel simulations are carried out employing both Wilson and the
O(a)-improved clover actions for the heavy quark. The discretization errors due
to the large $b$ quark mass are estimated in a systematic way with the aid of
the non-relativistic interpretation approach of El-Khadra, Kronfeld and
Mackenzie. As our best value from the quenched simulations at $\beta$=5.9, 6.1
and 6.3 we obtain $f_B$=163$\pm$16 MeV and $f_{B_s}$=175$\pm$18 MeV in the
continuum limit where the error includes both statistical and systematic
uncertainties.
