Radial velocities for 72 M dwarfs have been obtained having internal errors of about 0.1 km/s and external errors of about 0.4 km/s. Multiple velocity measurements of ten dMe stars have yielded a set of six which have no stellar companions, providing confirmation that the dMe phenomenon can occur in single stars. These single dMe stars have low space motions indicative of relative youth. Four stars from the entire survey were found to have double-line spectra and two were found to be single-line spectroscopic binaries of low amplitude. The zero point of the velocity scale is found to agree well with that of O. C. Wilson (1967) and differences are noted among other radial-velocity studies. Most of the stars in this study have velocities sufficiently well determined to constitute potential radial-velocity standards.