We report the discovery of an unusual X-ray burst from the direction of the Globular Cluster M28 using data acquired with the ASCA Observatory. The burst was recorded by all four ASCA telescopes and displays a fast (~ 70 ms) rise followed by an exponential decay (t = 7.5 s) and a steady afterglow which lasts between 800 - 3250 s. The image of the burst is consistent with an ASCA point source and is centered on quiescent X-ray emission from the core of M28. The burst temporal profile is similar to Type-I bursts emitted by accreting neutron stars of low mass X-ray binaries (LMXB). We argue that the burst arises from an LMXB that is located in the core of M28. The burst is unique in two ways: it is intrinsically sub-luminous, ~ 0.02 L_Edd and more importantly, originates from a source whose quiescent luminosity is fainter than that of the known cluster bursters by three orders of magnitude. We suggest that this burst is from a highly magnetized neutron star accreting at a low rate. These accreting systems may account for the mysterious low luminosity X-ray sources in globular clusters. ; Comment: 13 pages with 3 eps figures, LATEX, aastex, psfig. To appear in the Astrophysical Journal Letters