We present a new high-resolution N-body/SPH simulation of an encounter of two gas-rich disk galaxies which closely matches the morphology and kinematics of the interacting Antennae galaxies ( NGC 4038/39 ) . The simulation includes radiative cooling , star formation and feedback from SNII . The large-scale morphology and kinematics are determined by the internal structure and the orbit of the progenitor disks . The properties of the central region , in particular the starburst in the overlap region , only match the observations for a very short time interval ( \Delta t \approx 20 Myr ) after the second encounter . This indicates that the Antennae galaxies are in a special phase only about 40 { \thinspace Myr } after the second encounter and 50 { \thinspace Myr } before their final collision . This is the only phase in the simulation when a gas-rich overlap region between the nuclei is forming accompanied by enhanced star formation . The star formation rate as well as the recent star formation history in the central region agree well with observational estimates . For the first time this new model explains the distributed extra-nuclear star formation in the Antennae galaxies as a consequence of the recent second encounter . The proposed model predicts that the Antennae are in a later merger stage than the Mice ( NGC 4676 ) and would therefore lose their first place in the classical Toomre sequence .