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.
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