To study the properties of the interstellar medium in the prototypical merging system of the Antennae galaxies (NGC 4038 and NGC 4039), we have obtained CO(1-0), (2-1) and (3-2) line maps, as well as a map of the 870 micron continuum emission. Our results are analysed in conjunction with data from X-ray to radio wavelengths. In order to distinguish between exact coincidence and merely close correspondence of emission features, we compare the morphological structure of the different emission components at the highest available angular resolution. To constrain the physical state of the molecular gas, we apply models of photon dominated regions (PDRs) that allow us to fit CO and [CII] data, as well as other indicators of widespread PDRs in the Antennae system, particularly within the super giant molecular cloud (SGMC) complexes of the interaction region (IAR) between the two galaxies. The modeled clouds have cores with moderately high gas densities up to 4 10^4 / cm^3 and rather low kinetic temperatures <=25K). At present, all these clouds, including those near the galactic nuclei, show no signs of intense starburst activity. Thermal radio or mid-infrared emission are all observed to peak slightly offset from the molecular peaks. The total molecular gas mass of the Antennae system adds up to ~10^10 M_sun. In the vicinity of each galactic nucleus, the moleculargas mass, 1-2 10^9 M_sun, exceeds that of the Galactic centre region by a factorof almost 100. Furthermore, the gas does not seem to deviate much from the N_{H_2}/I_CO ratio typical of the disk of our Galaxy rather than our Galactic centre.
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