We present observations of NGC 4038/39 in the [ \ion C2 ] 158 \micron fine structure line taken with the MPE/UCB Far-infrared Imaging Fabry-Perot Interferometer ( FIFI ) on the KAO . A fully sampled map of the galaxy pair ( without the tidal tails ) at 55 \arcsec resolution has been obtained . The [ \ion C2 ] emission line is detected from the entire galaxy pair and peaks at the interaction zone . The total [ \ion C2 ] luminosity of the Antennae is L _ { [ CII ] } = 3.7 \times 10 ^ { 8 } L _ { \sun } , which is about 1 % of the far-infrared luminosity observed with IRAS . The main part of the [ \ion C2 ] emission probably arises from photodissociation regions ( PDRs ) , and a minor fraction may be emitted from \ion H2 regions . A small part of the [ \ion C2 ] emission comes from standard cold neutral medium ( CNM ) ; however , for high temperature ( T \sim 100 K ) and high density ( n _ { H } \sim 200 cm ^ { -3 } ) about one third of the observed [ \ion C2 ] emission may originate from CNM . From PDR models we derive densities of the order of \sim 10 ^ { 5 } cm ^ { -3 } and far-UV ( FUV ) intensities of 460 \chi _ { \circ } , 500 \chi _ { \circ } , and 240 \chi _ { \circ } for the PDRs in the interaction zone , NGC 4038 , and NGC 4039 , respectively . However , PDRs with densities of the order of \sim 10 ^ { 2 } cm ^ { -3 } and FUV intensities of the order of \sim 100 \chi _ { \circ } could also explain the observed [ \ion C2 ] emission . The minimum masses in the [ \ion C2 ] emitting regions in the interaction zone and the nuclei are a few \times 10 ^ { 7 } ~ { } M _ { \odot } . A comparison with single dish CO observations of the Antennae shows a [ \ion C2 ] to CO intensity ratio at the interaction zone 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 that at the nuclei of NGC 4038/39 . Therefore , no global starburst is taking place in the Antennae . [ \ion C2 ] emission arising partly from confined starburst regions and partly from surrounding quiescent clouds could explain the observed [ \ion C2 ] radiation at the interaction zone and the nuclei , though the star formation activity toward the nuclei is lower . 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 for the PDRs in the interaction zone and the nuclei .