By combining the first unquenched lattice computation of the B-meson binding energy and the two-loop contribution to the lattice HQET residual mass, we determine the <(MS)over bar> b-quark mass, (m) over bar(b)((m) over bar(b)). The inclusion of the two-loop corrections is essential to extract (m) over bar(b)((m) over bar(b)) with a precision of O(Lambda(QCD)(2)/m(b)), which is the uncertainty due to the renormalon singularities in the perturbative series of the residual mass. Our best estimate is (m) over bar(b)((m) over bar(b)) = (4.26 +/- 0.09) GeV, where we have combined the different errors in quadrature. A detailed discussion of the systematic errors contributing to the final number is presented. Our results have been obtained on a sample of 60 lattices of size 24(3) x 40 at beta = 5.6, using the Wilson action for light quarks and the lattice HQET for the b quark, at two values of the sea quark masses. The quark propagators have been computed using the unquenched links generated by the T chi L collaboration.