  A lattice calculation of the pseudoscalar decay constant of heavy-light
mesons is reported. Results are obtained (in the quenched approximation) from
lattices at $\beta=6.3$ through a procedure that interpolates between the
static approximation of Eichten and the conventional ("heavy" Wilson fermion)
method. The previously observed discrepancy between these two approaches has
been resolved: we find the scaling quantity $f\sqrt{M}$ to be significantly
smaller than previous calculations had indicated ({\it e.g.} at $\beta=6.0$);
in addition, we discuss a modification which is required in normalizing the
conventional amplitude to correct for large-$am$ lattice errors. This change
guarantees that $f\sqrt{M}$ will smoothly approach its value in the static
limit. From the numerical interpolation of the static and intermediate-mass
results, we find, in units of MeV, $f_B=187(10)\pm34\pm15$,
$f_{B_s}=207(9)\pm34\pm22$, $f_D=208(9)\pm35\pm12$ and
$f_{D_s}=230(7)\pm30\pm18$, where the first error is statistical and the second
two are estimates of systematics due to 1) fitting and large-$am$ effects and
2) scaling. In addition, using lattices at $\beta=6.3$, $\beta=6.0$ and
$\beta=5.7$ and extrapolating to the limit of zero lattice spacing, we have
computed $f_K/f_\pi = 1.08\pm.03\pm.08$ in the quenched approximation.
