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Precise near-infrared radial velocimetry enables efficient detection and transit verification of low-mass extrasolar planets orbiting M dwarf hosts, which are faint for visible-wavelength radial velocity surveys. The TripleSpec Exoplanet Discovery Instrument, or TEDI, is the combination of a variable-delay Michelson interferometer and a medium-resolution (R=2700) near-infrared spectrograph on the Palomar 200" Hale Telescope. We used TEDI to monitor GJ 699, a nearby mid-M dwarf, over 11 nights spread across 3 months. Analysis of 106 independent observations reveals a root-mean-square precision of less than 37 m/s for 5 minutes of integration time. This performance is within a factor of 2 of our expected photon-limited precision. We further decompose the residuals into a 33 m/s white noise component, and a 15 m/s systematic noise component, which we identify as likely due to contamination by telluric absorption lines. With further development this technique holds promise for broad implementation on medium-resolution near-infrared spectrographs to search for low-mass exoplanets orbiting M dwarfs, and to verify low-mass transit candidates., Comment: 55 pages and 13 figures in aastex format. Accepted by PASP