  We have investigated the Aharonov-Bohm effect in a one-dimensional
GaAs/GaAlAs ring at low magnetic fields. The oscillatory magnetoconductance of
these systems are for the first time systematically studied as a function of
density. We observe phase-shifts of $\pi$ in the magnetoconductance
oscillations, and halving of the fundamental $h/e$ period, as the density is
varied. Theoretically we find agreement with the experiment, by introducing an
asymmetry between the two arms of the ring.
