  We present a fully non-perturbative computation of the mass of the b-quark in
the quenched approximation. Our strategy starts from the matching of HQET to
QCD in a finite volume and finally relates the quark mass to the spin averaged
mass of the Bs meson in HQET. All steps include the terms of order Lambda^2/Mb.
Expanding on [1], we discuss the computation and renormalization of correlation
functions at order 1/Mb. With the strange quark mass fixed from the Kaon mass
and the QCD scale set through r0=0.5 fm, we obtain a renormalization group
invariant mass Mb = 6.758(86) GeV or mb(mb)= 4.347(48) GeV in the MS-bar
scheme. The uncertainty in the computed Lambda^2/Mb terms contributes little to
the total error and Lambda^3/Mb^2 terms are negligible. The strategy is
promising for full QCD as well as for other B-physics observables.
