[en] We have measured the microstability of the sloshing ions in the Tandem Mirror Experiment-Upgrade (TMX-U). This is the first experimental measurement of the microstability of a sloshing-ion distribution. We found that the sloshing ions were unstable to the drift cyclotron loss-cone instability. This instability was driven at the outer turning-point of the sloshing ions. We compare the experimental observations with the theoretical predictions for the drift cyclotron loss-cone instability. It is on the basis of these comparisons that we identify this instability as the drift cyclotron loss-cone intability. This instability affected the lifetime of the sloshing ions. We compare the observed scaling of the sloshing-ion lifetime with the amplitude of the instability to the results of a quasilinear theory where ion trapping is important. We show that this instability can be stabilized by two diferent cold-ion populations in TMX-U. We observe warm plasma stabilization of the instability by the cold ion endlosses. The amoun of stream necessary to stabilize the instability is consistent with the theory for the drift cyclotron loss-cone instability. We also show that warm ions trapped at the midplane can stabilize this instability, even though the instability is driven at a location removed from the midplane. This is the first experimental verification of the stabilizign effect of trapped ions for a sloshing-ion distribution. 125 refs., 33 figs