We study the late evolution of solar metallicity stars in the transition region between white dwarf formation and core collapse. This includes the super-asymptotic giant branch (super-AGB, SAGB) stars, which ignite carbon burning and form an oxygen-neon (ONe) core. SAGB star cores may grow to the Chandrasekhar mass because of continued H- and He-shell burning, ending as core-collapse supernovae. From stellar evolution models we find that the initial mass range for SAGB evolution is 7.5-9.25 M☉. We perform calculations with three different stellar evolution codes to judge the robustness of our results. The mass range significantly depends on the treatment of semiconvective mixing and convective overshooting. To consider the effect of a large number of thermal pulses, as expected in SAGB stars, we construct synthetic SAGB models that are calibrated through stellar evolution simulations. The synthetic model enables us to compute the evolution of the main properties of SAGB stars from the onset of thermal pulses until the core reaches the Chandrasekhar mass or is uncovered by the stellar wind. Thereby, we differentiate the stellar initial mass ranges that produce ONe WDs from that leading to electron-capture SNe. The latter is found to be 9.0-9.25 M☉ for our fiducial model, implying that electron-capture SNe would constitute about 4% of all SNe in the local universe. The error in this determination due to uncertainties in the third dredge-up efficiency and AGB mass-loss rate could lead to about a doubling of the number of electron-capture SNe, which provides a firm upper limit to their contribution to all supernovae of ~20%.