We present observations of NGC 4038/9 in the [C II] 158 μm fine-structure line taken with the MPE/UCB Far-infrared Imaging Fabry-Perot Interferometer (FIFI) on the Kuiper Airborne Observatory (KAO). A fully sampled map of the galaxy pair (without the tidal tails) at 55'' resolution has been obtained. The [C II] emission line is detected from the entire galaxy pair and peaks at the interaction zone. The total [C II] luminosity of the Antennae is L[C II] = 3.7 × 108 L☉, which is about 1% of the far-infrared luminosity observed with IRAS. The main part of the [C II] emission is probably produced by photodissociation regions (PDRs), and a minor fraction may be emitted from H II regions. A small part of the [C II] emission comes from a standard cold neutral medium (CNM); however, for high temperatures (T ~ 100 K) and high densities (nH ~ 200 cm-1) of the CNM, up to about one third of the observed [C II] emission may originate from CNM. From PDR models, we derive densities on the order of ~105 cm-3 and far-UV (FUV) intensities of 460χ0, 500χ0, and 240χ0 for the PDRs in the interaction zone, NGC 4038, and NGC 4039, respectively. However, PDRs with densities on the order of ~102 cm-3 and FUV intensities on the order of ~100χ0 could also explain the observed [C II] emission. The minimum masses in the [C II]-emitting regions in the interaction zone and the nuclei is a few ×107 M☉. A comparison with single-dish CO observations of the Antennae shows a [C II] to CO intensity ratio at the interaction zone that is a factor of 2.6 lower than usually observed in starburst galaxies, but still a factor of about 1.3 to 1.4 higher than at the nuclei of NGC 4038/9. Therefore, no global starburst is taking place in the Antennae. [C II] emission arising partly from confined starburst regions and partly from surrounding quiescent clouds could explain the observed [C II] radiation at the interaction zone and the nuclei. Accordingly, there are small confined regions with high star formation activity in the interaction zone and with a lower star formation activity in the nuclei. This supports the high density and high FUV intensity of the PDRs in the interaction zone and the nuclei.