  We report millimeter-wavelength observations of the Sunyaev-Zel'dovich (S-Z)
effect in two distant galaxy clusters. A relativistically correct analysis of
the S-Z data is combined with the results of X-ray observations to determine
the radial peculiar velocities v_r of the clusters. We observed Abell 2163
(z=.201) in three mm-wavelength bands centered at 2.1, 1.4, and 1.1 mm. We
report a significant detection of the thermal component of the S-Z effect seen
as both a decrement in the brightness of the CMB at 2.1 mm, and as an increment
at 1.1 mm. Including uncertainties due to the calibration of the instrument,
distribution and temperature of the IC gas, and astrophysical confusion, a
simultaneous fit to the data in all three bands gives v_r=+490 +1370/-880 km/s
at 68% confidence. We observed Abell 1689 (z=.181) in the 2.1 and 1.4 mm bands.
Including the same detailed accounting of uncertainty, a simultaneous fit to
the data in both bands gives v_r=+170 +815/-630 km/s. The limits on the
peculiar velocities of A2163 and A1689 correspond to deviations from the
uniform Hubble flow of <= 2-3%.
