We present trigonometric parallaxes of 64 stellar systems with proper motions between 0".5 yr(exp -1)and 1".0 yr(exp -1) from the ongoing Research Consortium On Nearby Stars parallax program at the Cerro Tololo Inter-American Observatory. All of the systems are south of decl. = +30, and 58 had no previous trigonometric parallaxes. In addition to parallaxes for the systems, we present proper motions, Johnson-Kron-Cousins VRI photometry, variability measurements, and spectral types. ...