A definitive determination of the progenitors of Type Ia supernovae (SNIa) has been a conundrum for decades. The single degenerate scenario -- a white dwarf (WD) in a semi-detached binary system accreting mass from its secondary -- is a plausible path; however, no simulation to date has shown that such an outcome is possible. In this study, we allowed a WD with a near Chandrasekhar mass of 1.4 M... to evolve over tens of thousands of nova cycles, accumulating mass secularly while undergoing periodic nova eruptions. We present the mass accretion limits within which an SNIa can possibly occur. The results showed, for each parameter combination within the permitted limits, tens of thousands of virtually identical nova cycles where the accreted mass exceeded the ejected mass, i.e. the WD grew slowly but steadily in mass. Finally, the WD became unstable, the maximal temperature rose by nearly two orders of magnitude, heavy element production was enhanced by orders of magnitude and the nuclear and neutrino luminosities became enormous. We also found that this mechanism leading to WD collapse is robust, with WDs in the range 1.0-1.38 M..., and an accretion rate of 5 x 10... M... yr..., all growing steadily in mass. These simulations of the onset of an SNIa event make observationally testable predictions about the light curves of pre-SN stars, and about the chemistry of SNIa ejecta. (ProQuest: ... denotes formulae/symbols omitted.)