We present optical spectroscopy of 16 star clusters in the merging galaxies NGC 4038/39 ("the Antennae") and supplement this data set with Hubble Space Telescope imaging. The age and metallicity of each cluster is derived through a comparison between the observed Balmer and metal line strengths with simple stellar population models. We then estimate extinctions and masses using the photometry. We find that all but three clusters have ages between similar to 3 and 200 Myr, consistent with the expected increase in the star formation rate (SFR) due to the merger. Most of the clusters have velocities in agreement with nearby molecular and Hi gas that has been previously shown to be rotating within the progenitor galaxies, hence star/cluster formation is still taking place within the galactic disks. However, three clusters have radial velocities that are inconsistent with being part of the rotating gas disks, which is surprising given their young (200-500 Myr) ages. Interestingly, we find a stellar association with the same colors (V - I) near one of these three clusters, suggesting that the cluster and association were formed concurrently and have remained spatially correlated. We find evidence for spatially distributed cluster formation throughout the duration of the merger. The impact of various assumptions about the SFR/cluster formation rate on the interpretation of the cluster age distribution is explored, and we do not find evidence for long-term "infant mortality" as has been previously suggested. Models of galaxy mergers that include a prescription for star formation can provide an overall good fit to the observed cluster age distribution.