Experiments on core centering in immiscible compound drops were conducted on the second United States Microgravity Laboratory Space Shuttle Mission (USML-2). It was observed that the inner and outer interfaces of a compound drop became concentric during capillary oscillations. These oscillations either were the freely decaying type, which occurred following the deployment of the drops, or were externally forced on the drop by modulating the positioning acoustic fields in the Drop Physics Module (DPM) experimental chamber. The results on core centering in liquid-liquid drops should stimulate theoretical studies in this field.