The M dwarf star L722-22 (LHS 1047, GJ 1005) was discovered to be a binary in 1979. Analysis of ground-based data indicated a mass near 0.06 M⊙ for the secondary star, well below the nominal stellar mass limit of 0.08 M⊙. The close, faint binary was near the limit for ground-based astrometry and was approved for Hubble Space Telescope Fine Guidance Sensor (FGS) observations in 1992. The relative orbital motion of the binary has been monitored using FGS "transfer" mode measurements. The trigonometric parallax and motion of the primary about the center of mass were determined from the FGS "position" mode observations. All possible background reference stars in the FGS field-of-view were used. The relative orbit and fractional masses have been determined with far higher precision and accuracy than possible with ground-based techniques for this close, faint binary. The FGS observations definitely eliminate the possibility that the secondary star is a candidate for having a substellar mass, and place its mass and lower mass error range well above the stellar mass limit. Masses of 0.179 and 0.112 M⊙ have been found for the two components, with formal random errors as low as 1.5%. The mass errors resulting from the correction from relative to absolute parallax are somewhat larger.