We present the evolution of helium stars in binary systems with a 1.4 M⊙ neutron-star companion which will produce double neutron-star binaries, i.e., systems with helium stars more massive than 2.5 M⊙. We found that mass transfer from helium star less massive than ~ 3.3 M⊙ will end up in a common-envelope phase. If the neutron star has enough time to complete the spiral-in before the core of the helium star collapses, the system will become a very tight double neutron-star binary (P ≈ 0d.01). More massive helium stars do not go through a dynamically-unstable mass transfer. The outcome of binaries with helium star in this range of mass is double-neutron star systems with period of 0d.1-1d, suggesting them to be the progenitor of the observed Galactic double neutron-star pulsars B 1913+16 and B 15344+12. Wide DNS pulsars like J 1518+4904 are produced from helium star-neutron star binaries which avoid Roche-lobe overflow. We are also able to distinguish the progenitors of Type Ib supernovae (as the high-mass helium stars or systems in wide orbits) from those of Type Ic supernovae (as the lower-mass helium stars or systems in close orbits).