We show that at any given time, the Galactocentric black hole Sgr A* is expected to be microlensing about 0.7 bulge stars if the threshold of detectability of the fainter image is K_thr=21, and about 4.6 sources if K_thr=23. The lensed images then provide a unique probe of the 20,000 stellar-mass black holes that are predicted to cluster within 0.7 pc of Sgr A*: if one of these black holes passes close to a microlensed image, it will give rise to a short (weeks long) microlensing event. We show that the mass and projected velocity of the black hole can both be measured unambiguously from such an event, if the lightcurve and the mean astrometric displacement can be measured. For K_thr=23 and moderate magnifications by Sgr A*, these events are expected only at a rate of 0.05 yr-1, so deeper observations would be required to see an event within the next decade. However, if highly magnified images of a star were found, the rate of events by the stellar black holes would be much higher.