We present mid-infrared spectroscopy of the prototypical interacting galaxies NGC 4038/39 (the 'Antennae') obtained with the ISO Short Wavelength Spectrometer (SWS). Our observations focus on the interaction zone where the two galaxies overlap, providing new constraints on the properties of the young and vigorous extranuclear starburst triggered by the recent interaction of the two galaxies. We use hydrogen recombination lines to derive an extinction of A(v) approximate to 70 to the starburst region, concluding that a model where emitting gas and absorbing dust are mixed best fits the data. We discuss the use of mid-infrared fine structure line ratios for determining the average effective temperature of the stellar ionizing radiation field, and derive an average temperature of 44000 K from the extinction-corrected [NeIII]15.6 mu m/[NeII]12.8 mu m line ratio. The observations are well described by starburst models for a recent starburst with an initial mass function extending up to 100 Mg.. Observations of the pure rotational (0-0) S(1) and S(2) lines of molecular hydrogen are used to constrain the fraction of molecular gas that is at temperatures of a few hundred Kelvin.