We investigate the spin of the ground state of a geometrically confined many-electron system. For atoms, shell structure simplifies this problem—the spin is prescribed by the well-known Hund’s rule. In contrast, quantum dots provide a controllable setting for studying the interplay of quantum interference and electron-electron interactions in general cases. In a generic confining potential, the shell-structure argument suggests a singlet ground state for an even number of electrons. The interaction among the electrons produces, however, accidental occurrences of spin-triplet ground states, even for weak interaction, a limit which we analyze explicitly. Variation of an external parameter causes sudden switching between these states and hence a kink in the conductance. Experimental study of these kinks would yield the exchange energy for the “chaotic electron gas”. © 2000 The American Physical Society.