Nuclear double-$\ensuremath{\beta}$ decay is discussed on the assumptions (1) that the process, treated as second-order weak, occurs predominantly without neutrino emission and so violates lepton convervation, and (2) that the matrix element for the process is mainly due to the no-neutrino double-$\ensuremath{\beta}$ decay of an isospin-$\frac{3}{2}$ nucleon resonance present, albeit with a small (\ensuremath{\cong}1%) probability, in the parent or daughter nucleus. The theoretical lifetimes obtained on this basis are compared with the corresponding experimental lifetimes; the various lifetime ratios are correctly predicted, and a limit \ensuremath{\cong}${10}^{\ensuremath{-}4}$ is imposed on the relevant "lepton-non-conservation" parameter. A discussion is also given of the convergence problems associated with the matrix elements for second-order weak processes in general. In addition it is shown that in spite of the presence of virtual-neutrino closed loops, divergences never arise in the matrix elements for no-neutrino double-$\ensuremath{\beta}$ decay if the basic nuclear constituents (i.e., quarks or nucleons) have isospin not exceeding \textonehalf{} and mean separation greater than zero.