  We demonstrate how HQET and the Step Scaling Method for B-physics, pioneered
by the Tor Vergata group, can be combined to reach a further improved
precision. The observables considered are the mass of the b-quark and the
B_s-meson decay constant. The demonstration is carried out in quenched lattice
QCD. We start from a small volume, where one can use a standard O(a)-improved
relativistic action for the b-quark, and compute two step scaling functions
which relate the observables to the large volume ones. In all steps we
extrapolate to the continuum limit, separately in HQET and in QCD for masses
below m_b. The physical point m_b is then reached by an interpolation of the
continuum results in 1/m. The essential, expected and verified, feature is that
the step scaling fuctions have a weak mass-dependence resulting in an easy
interpolation to the physical point. With r_0=0.5fm and the experimental B_s
and K masses as input, we find F_Bs=191(6)MeV and the renormalization group
invariant mass M_b=6.88(10)GeV, translating into m_b(m_b)=4.42(6)GeV in the
MSbar scheme. This approach seems very promising for full QCD.
