  We have probed the magnetic field dependence of the electron phase coherence
time $\tau_\phi$ by measuring the Aharonov-Bohm conductance oscillations of
mesoscopic Cu rings. Whereas $\tau_\phi$ determined from the low-field
magnetoresistance saturates below 1 K, the amplitude of Aharonov-Bohm $h/e$
oscillations increases strongly on a magnetic field scale proportional to the
temperature. This provides strong evidence that a likely explanation for the
frequently observed saturation of $\tau_\phi$ at low temperature in weakly
disordered metallic thin films is the presence of extremely dilute magnetic
impurities.
