The Multichannel Astrometric Photometer (MAP) has been used to determine parallaxes of stars in the region centered on the Mira variable R Leonis. R Leonis, with a parallax of 8.3 +/- 1.0 mas (120 +/- 15 pc), is significantly closer than most recent estimates. At visual wavelengths, its mean physical diameter is 612 +/- 96 times that of the Sun. The photocentric motion of R Leonis is not suggestive of either optical or orbital companions, excluding possible secondary components over a wide range of magnitudes as well as orbital periods of up to approximately one decade. Nineteen Leonis, an occultation-speckle binary with a well-determined orbit and a reference star in the R Leonis field, has a distance of 92.6 +/- 12 pc. The 15.2 yr period binary exhibits a slight perturbation during the 4 yr of MAP observations. The similar primary and secondary of this A7 Vn system are 2.41 +/- 0.96 and 2.31 +/- 0.92 solar masses, respectively.