  We have calculated the evolution of 60 model binary systems consisting of
helium stars in the mass range of M_He= 2.5-6Msun with a 1.4Msun neutron star
companion to investigate the formation of double neutron star systems.Orbital
periods ranging from 0.09 to 2 days are considered, corresponding to Roche lobe
overflow starting from the helium main sequence to after the ignition of carbon
burning in the core. We have also examined the evolution into a common envelope
phase via secular instability, delayed dynamical instability, and the
consequence of matter filling the neutron star's Roche lobe. The survival of
some close He-star neutron-star binaries through the last mass transfer episode
(either dynamically stable or unstable mass transfer phase) leads to the
formation of extremely short-period double neutron star systems (with
P<~0.1days). In addition, we find that systems throughout the entire calculated
mass range can evolve into a common envelope phase, depending on the orbital
period at the onset of mass transfer. The critical orbital period below which
common envelope evolution occurs generally increases with M_He. In addition, a
common envelope phase may occur during a short time for systems characterized
by orbital periods of 0.1-0.5 days at low He-star masses (~ 2.6-3.3Msun).
  The existence of a short-period population of double neutron stars increases
the predicted detection rate of inspiral events by ground-based
gravitational-wave detectors and impacts their merger location in host galaxies
and their possible role as gamma-ray burst progenitors. We use a set of
population synthesis calculations and investigate the implications of the
mass-transfer results for the orbital properties of DNS populations.
