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We present new VLA C+D array H I observations and optical and near-infrared imaging of the well-known interacting system NGC 4038/39, ``The Antennae.'' At low spatial resolution (∼40"), the radio data reach a limiting column density of ∼10<sup>19</sup> cm<sup>–2</sup> (2.5 σ), providing significantly deeper mapping of the tidal features than afforded by earlier H I maps. At relatively high spatial resolution (∼10"), the radio data reveal a wealth of gaseous substructure both within the main bodies of the galaxies and along the tidal tails. In agreement with previous H I studies, we find that the northern tail has H I along its outer length but none along its base. We suggest that the H I at the base of this tail has been ionized by massive stars in the disk of NGC 4038. The gas in the southern tail has a bifurcated structure, with one filament lying along the optical tail and another running parallel to it but with no optical counterpart. The two filaments join just before the location of several star-forming regions near the end of the tail. The H I velocity field at the end of the tail is dominated by strong velocity gradients, which suggests that at this location the tail is bending away from us. We delineate and examine two regions within the tail previously identified as possible sites of a so-called tidal dwarf galaxy condensing out of the expanding tidal material. The tail velocity gradients mask any clear kinematic signature of a self-gravitating condensation in this region. A dynamical analysis suggests that there is not enough mass in gas alone for either of these regions to be self-gravitating. Conversely, if they are bound they require a significant contribution to their dynamical mass from evolved stars or dark matter. Even if there are no distinct dynamical tidal entities, it is clear that there is a unique concentration of gas, stars, and star-forming regions within the southern tail: the H I channel maps show clear evidence for a significant condensation near the tail star-forming regions, with the single-channel H I column densities higher than anywhere else in the system, including within the main disks. Finally, the data reveal H I emission associated with the edge-on ``superthin'' Scd galaxy ESO 572-G045, which lies just beyond the southern tidal tail of The Antennae, showing it to be a companion system.