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The M dwarf binary, Wolf 1062 (Gliese 748), has been observed with the Hubble Space Telescope (HST) Fine Guidance Sensor 3 in the transfer function scan mode to determine the apparent orbit. This is the first orbit defined fully and exclusively with HST, and is the most accurate definitive orbit for any resolved, noneclipsing system. The orbital period is 2.4490 ± 0.0119 yr and the semimajor axis is 01470 ± 00007—both quantities are now known to better than 1%. Using the weighted mean of seven parallax measurements and these HST data, we find the system mass to be 0.543 ± 0.031 M, where the error of 6% is due almost entirely to the parallax error. An estimated fractional mass from the infrared brightness ratio and infrared mass-luminosity relation yields a mass for the primary of 0.37 M, and the secondary falls in the regime of very low mass stars, with a mass of only 0.17 M.