  The evolution of helium stars with masses of 1.5 - 6.7 M_sun in binary
systems with a 1.4 M_sun neutron-star companion is presented. Such systems are
assumed to be the remnants of Be/X-ray binaries with B-star masses in the range
of 8 - 20 M_sun which underwent a case B or case C mass transfer and survived
the common-envelope and spiral-in process. The orbital period is chosen such
that the helium star fills its Roche lobe before the ignition of carbon in the
centre. We distinguish case BA (in which mass transfer is initiated during
helium core burning) from case BB (onset of Roche-lobe overflow occurs after
helium core burning is terminated, but before the ignition of carbon). We found
that the remnants of case BA mass transfer from 1.5 - 2.9 M_sun helium stars
are heavy CO white dwarfs. This implies that a star initially as massive as 12
M_sun is able to become a white dwarf. CO white dwarfs are also produced from
case BB mass transfer from 1.5 - 1.8 M_sun helium stars, while ONe white dwarfs
are formed from 2.1 - 2.5 M_sun helium stars. Case BB mass transfer from
more-massive helium stars with a neutron-star companion will produce a double
neutron-star binary. We are 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). Finally, we derive a "zone of avoidance" in the helium star mass vs.
initial orbital period diagram for producing neutron stars from helium stars.
