  We report simultaneous observations at 1.2 and 2 mm, with a double channel
photometer on the SEST Telescope, of the X-ray cluster RXJ0658-5557 in search
for the Sunyaev-Zel'dovich (S-Z). The S-Z data were analyzed using the
relativistically correct expression for the Comptonization parameter and we
find from the detected decrement (2.60 +/- 0.79) ~ 10^{-4}, which is consistent
with that computed using the X-ray (ROSAT and ASCA) observations. The
uncertainty includes contributions due to statistical uncertainty in the
detection and systematics and calibration. The 1.2 {mm} channel data alone
gives rise to a larger Comptonization parameter and this result is discussed in
terms of contamination from foreground sources and/or dust in the cluster or
from a possible systematic effect. We then make use of the combined analysis of
the ROSAT and ASCA X-ray satellite observations to determine an isothermal
model for the S-Z surface brightness. Since the cluster is asymmetrical and
probably in a merging process, models are only approximate. The associated
uncertainty can, however, be estimated by exploring a set of alternative
models. We then find as the global uncertainty on the Comptonization parameter
a factor of 1.3. Combining the S-Z and the X-ray measurements, we determine a
value for the Hubble constant. The 2 mm data are consistent with H_0(q_0 =
1/2)= 53^{+38}_{-28} km/s Mpc^{-1}, where the uncertainty is dominated by the
uncertainty in models of the X-ray plasma halo.
