We present Hubble Space Telescope data of the low-reddening Sagittarius window in the Galactic bulge. The Sagittarius Window Eclipsing Extrasolar Planet Search field (~3'x3'), together with three more Advanced Camera for Surveys and eight Wide-Field Camera 3 fields, were observed in the F606W and F814W filters, approximately every two weeks for 2 yr, with the principal aim of detecting a hidden population of isolated black holes and neutron stars through astrometric microlensing. Proper motions were measured with an accuracy of ~0.1 mas/yr (~4 km/s) at F606W ~25.5 mag, and better than ~0.5 mas/yr (~20 km/s) at F606W ~28 mag, in both axes. Proper-motion measurements allowed us to separate disk and bulge stars and obtain a clean bulge color-magnitude diagram. We then identified for the first time a white dwarf (WD) cooling sequence in the Galactic bulge, together with a dozen candidate extreme horizontal branch stars. The comparison between theory and observations shows that a substantial fraction of the WDs (~30%) are systematically redder than the cooling tracks for CO-core H-rich and He-rich envelope WDs. This evidence would suggest the presence of a significant number of low-mass WDs and WD-main-sequence binaries in the bulge. This hypothesis is further supported by the finding of two dwarf novae in outburst, two short-period (P <~ 1 day) ellipsoidal variables, and a few candidate cataclysmic variables in the same field.