  We present a high statistics study of the leptonic decay constant $f_P$ of
heavy pseudoscalar mesons using propagating heavy Wilson quarks within the
quenched approximation, on lattices covering sizes from about 0.7~fm to 2~fm.
Varying $\beta$ between 5.74 and 6.26 we observe a sizeable $a$ dependence of
$f_P$ when one uses the quark field normalization that was suggested by
Kronfeld and Mackenzie, compared with the weaker dependence observed for the
standard relativistic norm. The two schemes come into agreement when one
extrapolates to $a \rightarrow 0$. The extrapolations needed to reach the
continuum quantity $f_B$ introduce large errors and lead to the value
$f_B=0.18(5)$~GeV in the quenched approximation. This suggests that much more
effort will be needed to obtain an accurate lattice prediction for $f_B$.
