The evolution of primary components of initially close binary systems has been followed assuming specific initial conditions such as to have the formation of an O-Ne white dwarf at the end of a nonconservative evolution of the system parameters. More specifically, the evolution of a 10.5 Msun, Z = Zsun, and Y = 0.27 star model has been traced through the two major episodes of mass transfer in order to make a direct comparison with the previous computations of Iben & Tutukov, which we reproduce quite well. In this case, mass transfer finishes at the beginning of the C-burning phase. Analogous experiments have been performed with both a 10 Msun and a 10.5 Msun (Z = 2 × Zsun and Y = 0.32) star models. In these cases, the second mass-transfer episode continues also during the carbon-burning phase. In all cases considered, the carbon-burning phase has been traced to the onset of C-burning in an external shell (after central carbon exhaustion) and, in one case, up to the epoch when any nuclear burning has been completely extinguished.For comparison, we have evolved constant-mass models over a range of masses. In addition to the expected property that in envelope-deprived star models carbon ignition starts off center (whereas in the parent constant mass models it would start at the center), we find that some evolutionary properties of our off-center carbon-burning models differ from those of previously published models. However, these differences, which may be ascribed to different input physics, do not substantially alter the final outcome, except for the fact that in very low mass O-Ne remnants a small amount of carbon is left unburned at the center. It is found that the mass of the outcoming O-Ne white dwarf increases, increasing the initial metallicity of the components, an occurrence which, however, is mainly determined by the fact that for more metal rich systems we also adopt a higher initial He abundance. Finally, we discuss how present results affect current scenarios concerning the outcome of mass accretion onto a O-Ne white dwarf: collapse to a neutron star or explosion leaving no compact remnant.