We present observations of NGC 4038/9 in the [C II] 158 mu m 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 x 10(8) 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 similar to 100 K) and high densities (n(H) similar to 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 similar to 10(5) cm(-3) and far-UV (FUV) intensities of 460 chi(0), 500 chi(0), and 240 chi(0) for the PDRs in the interaction zone, NGC 4038, and NGC 4039, respectively. However, PDRs with densities on the order of similar to 10(2) cm(-3) and FUV intensities on the order of similar to 100 chi(0) 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 x 10(7) 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.