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We derive a new equation of state (EoS) for neutron stars (NS) from the outer crust to the core based on modern microscopic calculations using the Argonne v18 potential plus three-body forces computed with the Urbana model. To deal with the inhomogeneous structures of matter in the NS crust, we use a recent nuclear energy density functional that is directly based on the same microscopic calculations, and which is able to reproduce the ground-state properties of nuclei along the periodic table. The EoS of the outer crust requires the masses of neutron-rich nuclei, which are obtained through Hartree-Fock-Bogoliubov calculations with the new functional when they are unknown experimentally. To compute the inner crust, Thomas-Fermi calculations in Wigner-Seitz cells are performed with the same functional. Existence of nuclear pasta is predicted in a range of average baryon densities between ≃0.067 fm-3 and ≃0.0825 fm-3, where the transition to the core takes place. The NS core is computed from the new nuclear EoS assuming non-exotic constituents (core of npeμ matter). In each region of the star, we discuss the comparison of the new EoS with previous EoSs for the complete NS structure, widely used in astrophysical calculations. The new microscopically derived EoS fulfills at the same time a NS maximum mass of 2 M⊙ with a radius of 10 km, and a 1.5 M⊙ NS with a radius of 11.6 km.