We report the detection with the proportional counter array (PCA) onboard the Rossi X-Ray Timing Explorer (RXTE) of 589 Hz oscillations during three type I X-ray bursts from a low-mass X-ray binary located in the Galactic center region. The bursts, which occurred on 1996 May 15.814, June 4.612, and June 19.414 UT, were observed serendipitously during routine monitoring observations of GRO J1744-28 by RXTE. The strongest pulsations were observed from the May 15 burst, reaching an amplitude of 18% of the >8 keV flux in this burst. The pulsation amplitudes in each burst are a strong function of photon energy since no signals were detected in the 1-8 keV band, but there were strong detections above 8 keV. The evolution of the X-ray spectrum through each burst is consistent with modest radius expansion. The pulsations are confined to the postcontraction portion of each burst profile, consistent with the oscillations recently reported in a burst from KS 1731-26 and also some bursts from 4U 1728-34. The location of the burster in the 1° (FWHM) PCA field of view is constrained by comparing the count rates in the five independent detectors of the PCA. This analysis strongly excludes GRO J1744-28 as the source of these bursts but does not yet allow a definitive identification of the source with any known burster in the field of view. However, the derived position strongly overlaps that of MXB 1743-29, identifying this source as a strong candidate for the 589 Hz burster. We argue that the observed oscillations are consistent with rotational modulation of the X-ray brightness.