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 MSbar b-quark mass, m̅_b(m̅_b). The inclusion of the two-loop corrections is essential to extract m̅_b(m̅_b) with a precision of O(Λ²_QCD/m_b), which is the uncertainty due to the renormalon singularities in the perturbative series of the residual mass. Our best estimate is m̅_b(m̅_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 24³ × 40 at β = 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 TχL collaboration.