In May, August, and November 1995, Hubble Space Telescope (HST) observations during Saturn's ring-plane crossings allowed us to view saturnian satellites normally hidden to Earth-based observers in the glare of the rings. New measurements of Janus, Epimetheus, Prometheus, and Pandora have been combined to form revised orbital solutions using all three HST data sets. These measurements and orbit fits are presented, as well as similar fits for the brighter satellites Mimas, Tethys, Enceladus, Dione, and Rhea. Observations of the Lagrangian satellites Telesto, Calypso, and Helene are also reported. While most satellites were found to be close to their expected positions based on previous orbital solutions (Nicholson et al. 1992, Icarus 100, 464-484; Jacobson 1996, Bull. Amer Astron. Soc. 28, 1185; and Harper and Taylor 1993, Astron. Astrophys. 268, 326-349). Prometheus lagged behind its predicted longitude by 18.85 degrees +/- 0.04 degrees. A systematic drift in Pandora's longitude of -1.85 degrees relative to the Voyager ephemeris was observed between May and November. The new data on the coorbital satellites Janus and Epimetheus have resulted in a revised mass for Janus, similar to6% smaller than the previous value (Jacobson 1995, Bull. Amer. Astron. Soc. 27, 1202). Subtraction of light from the edge-on rings has led to additional detections of objects S/1995-S1 and S3 (Bosh and Rivkin 1995, Science 272, 518-521) in the May data, and S/1995-S5, S6, S7 (Nicholson et al. 1995, Bull. Amer Astron. Soc. 27, 1202) and S/1995-S9 (Roddier et al. 1996) in the August images. S1 is identified with Atlas but leads its predicted position by similar to 25 degrees. S3 has an orbit consistent with that of the narrow F ring, but S5, S6, S7, and S9 now appear to orbit similar to 530-950 kin interior to this ring. S7 and S9 may even be coorbital with Prometheus. An object in the May images, possibly corresponding to S7, is also found to lie very close to Prometheus' orbit (similar to 800 kin interior to the F ring), but no convincing detections of an analogous object in this orbit at this longitude were made in the November data set. In addition to these previously reported objects, eight new objects were identified during multiple HST visits in May and August; all have orbits consistent with that of the F ring and are likely to have been short-lived clumps within this ring, similar to those seen in Voyager images (C) 2001 Academic Press.