  We report the results of the analysis of the extended soft emission
discovered in the Chandra ACIS pointing at the merging system NGC 4038/39 (the
Antennae). We present a `multi-color' X-ray image that suggests both extensive
absorption by the dust in this system, peaking in the contact region, as well
as variations in the temperature of different emitting regions of the hot
interstellar medium (ISM). Spectral fits to multi-component thermal emission
models confirm this picture and give a first evaluation of the parameters of
the hot plasma. We compare the diffuse X-ray emission with radio continuum
(6cm), HI, CO, and H$\alpha$ images to take a first look at the multi-phase ISM
of the Antennae galaxies. We find that the hot (X-ray) and cold (CO) gas have
comparable thermal pressures in the two nuclear regions. We also conclude that
the displacement between the peak of the diffuse X-ray emission in the north of
the galaxy system, towards the inner regions of the northern spiral arm (as
defined by H$\alpha$, radio continuum and HI), could result from ram pressure
of infalling HI clouds.
