We have begun to observe radial velocities of stars with an optical spectrometer designed for unusually high accuracy. Light from a star image in the focal plane of a telescope is fed to the entrance aperture of the spectrometer by a single optical fiber. Wavelengths are calibrated by transmission of collimated light through a tilt-tunable Fabry-Perot interferometer. The scrambling of incident light rays by the optical fiber and the intrinsic stability of the Fabry-Perot etalon provide immunity to the sources of systematic errors that plague conventional radial velocity meters. The spectrum is dispersed by an echelle grating crossed with another plane reflection grating. Several echelle orders in the vicinity of 4250-4600 Å are imaged in a two-dimensional format on a charge-coupled (CCD) array of detectors. About 350 distinct points on the profile of the stellar spectrum are sampled by successive orders of interferometrie transmission through the etalon. In the vicinity of 4300 Å each interference order is 47 milliangstroms wide and the sample points are 0.64 Å apart, resulting in distinct , widely-spaced monochromatic images of the entrance aperture to be formed in the focal plane of the camera. Changes in Doppler shift cause changes in the relative intensities of these images, according to the slope of the spectral profile at each point sampled. The instrument is being operated as a null-measurement accelerometer, sensitive only to changes in radial velocity, which simplifies operation and enhances sensitivity. With an argon-filled, iron hollow cathode emission line lamp, the interferometer can be calibrated to two parts in 100 million; this corresponds to ± 6 meters/sec in Doppler shift. Calibrations of the interferometer show variations of ± 27 meters/sec on a time scale of months; observations of stars are corrected for such changes. The internal repeatability of observations of the differential Doppler shift of light from the integrated disk of the Sun is ± 6 meters/sec. The corresponding result from about 70 observations of Arcturus (Kl IIIb; B=1.19) is ± 40 meters/sec internal repeatability for each exposure of 20 square-meter seconds. The external repeatability (day-to-day differential accuracy) of nightly averages of stellar observations is ± 20 meters/second. Since the internal precision on the sun and the argon lamp is much better than it is with short exposures on Arcturus, the quality of our observations of stars is limited by the rate of detected photons. This justifies averaging a number of short exposures of a star to approach “laboratory” precision.