  We derive a new equation of state (EoS) for neutron stars (NS) from the outer
crust to the core based on modern microscopic Brueckner-Hartree-Fock (BHF)
calculations using the Argonne $v_{18}$ potential plus three-body forces
computed with the Urbana model. To deal with the inhomogeneous structures of
matter in the NS crust, we use the recent Barcelona-Catania-Paris-Madrid (BCPM)
nuclear energy density functional that is directly based on the same
microscopic BHF 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 BCPM 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
$\simeq$0.067 fm$^{-3}$ and $\simeq$0.0825 fm$^{-3}$, where the transition to
the core takes place. The NS core is computed from the nuclear EoS of the BHF
calculation assuming non-exotic constituents (core of $npe\mu$ matter). In each
region of the star, we discuss the comparison of the new EoS with previous
EoSes for the complete NS structure, in particular, with the Lattimer-Swesty
EoS and with the Shen et al. EoS widely used in astrophysical calculations. The
new microscopically derived EoS fulfills at the same time a NS maximum mass of
2~$M_\odot$ with a radius of 10 km, and a 1.5~$M_\odot$ NS with a radius of
11.7 km.
