  The buildup of low-energy electrons has been shown to affect the performance
of a wide variety of particle accelerators. Of particular concern is the
persistence of the cloud between beam bunch passages, which can impose
limitations on the stability of operation at high beam current. We have
obtained measurements of long-lived electron clouds trapped in the field of a
quadrupole magnet in a positron storage ring, with lifetimes much longer than
the revolution period. Based on modeling, we estimate that about 7% of the
electrons in the cloud generated by a 20-bunch train of 5.3 GeV positrons with
16-ns spacing and $1.3x10^{11}$ population survive longer than 2.3 $\mu$s in a
quadrupole field of gradient 7.4 T/m. We have observed a non-monotonic
dependence of the trapping effect on the bunch spacing. The effect of a witness
bunch on the measured signal provides direct evidence for the existence of
trapped electrons. The witness bunch is also observed to clear the cloud,
demonstrating its effectiveness as a mitigation technique.
