We are measuring temporal changes of the Doppler shifts of stars with the aid of a Fabry-Perot interferometer. A combination of design features permits unusually good precision and accuracy. First, an optical fiber feeds starlight from the telescope focus to the object plane of the collimator of a cross-dispersed echelle spectrograph. The length and scrambling properties of the fiber isolate the interferometer, and thereby the velocity metric, from environmental conditions and the effects of guiding and seeing. The etalon, in parallel light between the collimator and the echelle grating, is tuned by tilting to allow the transmitted orders of constructive interference to fall on the desired wavelengths. The spectral resolution and the absolute wavelength metric are defined by the interferometer, not by the placement of the spectrum on the detector. This avoidance of mapping CCD pixel position into radial velocity is a fundamental feature that enhances accuracy. Finally, the spectrograph spatially separates the interference orders onto a CCD. Changes of the velocity of a star are sensed by detecting changes in the relative intensities of those interference orders that lie on the steep slopes of line profiles. The parameters of the interferometer are measured frequently by observing emission lines from an iron-argon hollow cathode lamp. Years of use show that with such calibration, observations of stars can be reduced to the same velocity reference with an accuracy of ±10 m/s. This corresponds to ±33 ppb in the effective spacing of the etalon plates.