We report on high dispersion spectroscopy of the Ha absorption line of the cool DA white dwarf G 29 -38. This is the star for which a recently detected infrared excess has been suggested to be due to a possible brown dwarf companion by Zuckerman and Becklin (1986, 1987). Three echelle spectra obtained at the Multiple Mirror Telescope and at the Kitt Peak Mayall 4m telescope in 1987 December show no evidence for radial velocity variations larger than -'1.1 ± 8.7 km s -1 and are used to derive a weighted heliocentric radial velocity Vr = 33.7 ± 4.3 km s -1 for the white dwarf. No emission component from the hypothesized secondary star is detected. These negative results do not constitute strong evidence against the companion hypothesis, since the expected orbital velocity of the white dwarf component could be quite small, and the companion's line emission could be too faint to be detected. However, the observation of a sharp absorption line core restricts the possible rotation of the white dwarf to < 40 km s -1 and ensures that any surface magnetic field has a strength < 105 gauss. These results make it unlikely that the DA white dwarf has previously been in a cataclysmic variable accretion phase.