  We study the amplitude of mesoscopic Aharonov-Bohm oscillations in
quasi-one-dimensional (Q1D) diffusive rings. We consider first the
low-temperature limit of a fully coherent sample. The variance of oscillation
harmonics is calculated as a function of the length of the leads attaching the
ring to reservoirs. We further analyze the regime of relatively high
temperatures, when the dephasing due to electron-electron interaction
suppresses substantially the oscillations. We show that the dephasing length
L_phi^AB governing the damping factor exp(-2pi R /L_phi^AB) of the oscillations
is parametrically different from the common dephasing length for the Q1D
geometry. This is due to the fact that the dephasing is governed by energy
transfers determined by the ring circumference 2pi R, making L_phi^AB
R-dependent.
