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We present a fully non-perturbative computation of the mass of the b-quark inthe quenched approximation. Our strategy starts from the matching of HQET toQCD in a finite volume and finally relates the quark mass to the spin averagedmass 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 correlationfunctions at order 1/Mb. With the strange quark mass fixed from the Kaon massand the QCD scale set through r0=0.5 fm, we obtain a renormalization groupinvariant mass Mb = 6.758(86) GeV or mb(mb)= 4.347(48) GeV in the MS-barscheme. The uncertainty in the computed Lambda^2/Mb terms contributes little tothe total error and Lambda^3/Mb^2 terms are negligible. The strategy ispromising for full QCD as well as for other B-physics observables.Comment: 34 pages, 9 figures, 12 tables references added and minor correction

Heavy quark effective theory computation of the mass of the bottom quark