  We present a new determination of the $B_s$ leptonic decay constant from
lattice QCD simulations that use gluon configurations from MILC and a highly
improved discretization of the relativistic quark action for both valence
quarks. Our result, $f_{B_s} = 0.225(4)$\,GeV, is almost three times more
accurate than previous determinations. We analyze the dependence of the decay
constant on the heavy quark's mass and obtain the first empirical evidence for
the leading $1/\sqrt{m_h}$ dependence predicted by Heavy Quark Effective Theory
(HQET). As a check, we use our analysis technique to calculate the
$m_{B_s}-m_{\eta_b}/2$ mass difference. Our result agrees with experiment to
within errors of $11\,\mathrm{MeV}$ (better than 2%). We discuss how to extend
our analysis to other quantities in $B_s$ and $B$ physics, making 2%-precision
possible for the first time.
