We have investigated the Aharonov-Bohm effect in a one-dimensional (Formula presented) ring at low-magnetic fields. The oscillatory magnetoconductance of these systems is systematically studied as a function of density. We observe phase shifts of π in the magnetoconductance oscillations, and halving of the fundamental (Formula presented) period, as the density is varied. Theoretically we find agreement with the experiment, by introducing an asymmetry between the two arms of the ring.