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\noindent Paczy\'nski and Zi\'o\l kowski (1968) suggested that stars on the asymptotic giant branch (AGB) may lose their envelopes when the binding energy of their envelopes becomes positive. Adopting this criterion and using the most recent available opacities, we have performed a series of stellar evolution calculations to determine the relation between the initial main-sequence and the final white-dwarf mass for single stars. The resulting relationship is consistent, within the observational error, with the observational relationship by Weidemann and Koester (1983). Furthermore, convolving our theoretical relation with the observed initial mass function, we obtain a theoretical, essentially parameter-free distribution of white dwarf masses. This distribution has a sharp peak at $0.61 \pm 0.02\,M_{\odot}$ with a long non-Gaussian tail to higher masses and possibly a smaller peak at $0.46\pm 0.01\,M_{\odot}$ (for single helium white dwarfs). \par\medskip Our theoretical distribution appears to be in remarkably good agreement with various observational determinations of the masses of the degenerate cores in planetary nebulae. In addition, the distribution is not very sensitive to the assumed wind mass-loss rate or the adopted mixing-length parameter. However, it is sensitive to the metallicity of the population. For the same initial mass, the final white-dwarf mass is $5\,$--$\,10\,$\% larger for population II stars with $Z=0.001$. These results are not much changed when the effects of binary evolution are included (except that the distribution of helium white dwarfs becomes more important and much broader). \par\medskip Finally, we discuss the implications of this model for using planetary nebulae as distance indicators and possible envelope ejection mechanisms (dynamical ejection, a pulsationally driven superwind, etc.). \par\bigskip\noindent Paczy\'nski, B. \& Zi\'o\l kowski, J., 1968, Acta Astron., 18, 255.\\ Weidemann, V. \& Koester, D., 1983, A\&A, 121, 77.
