Quantum mechanical experiments in ring geometries have long fascinated physicists. Open rings connected to leads, for example, allow the observation of the Aharonov-Bohm effect(1), one of the best examples of quantum mechanical phase coherence(2,3). The phase coherence of electrons travelling through a quantum dot embedded in one arm of an open ring has also been demonstrated(4). The energy spectra of closed rings(5) have only recently been studied by optical spectroscopy(6,7). The prediction that they allow persistent current(8) has been explored in various experiments(9-11). Here we report magnetotransport experiments on closed rings in the Coulomb blockade regime(12). Our experiments show that a microscopic understanding of energy levels, so far limited to few-electron quantum dots(13), can be extended to a many-electron system. A semiclassical interpretation of our results indicates that electron motion in the rings is governed by regular rather than chaotic motion, an unexplored regime in many-electron quantum dots. This opens a way to experiments where even more complex structures can be investigated at a quantum mechanical level.