The ultraluminous infrared galaxy Arp 220 has been observed at 0farcs5 resolution in CO (2-1) and 1 mm continuum using the newly expanded Owens Valley Millimeter Array. The CO and continuum peaks at the double nuclei and the surrounding molecular gas disk are clearly resolved. We find steep velocity gradients across each nucleus (ΔV~500 km s-1 within r=0farcs3) whose directions are not aligned with each other and with that of the outer gas disk. We conclude that the double nuclei have their own gas disks (r~100 pc), are counterrotating with respect to each other, and are embedded in the outer gas disk (r~1 kpc) rotating around the dynamical center of the system. The masses of each nucleus are Mdyngtrsim2×109 Msun, based on the CO kinematics. Although there is no evidence of an old stellar population in the optical or near-infrared spectroscopy of the nuclei (probably owing to the much brighter young population), it seems likely that these nuclei were "seeded" from the premerger nuclei in view of their counterrotating gas kinematics. The gas disks probably constitute a significant fraction (~50%) of the mass in each nucleus. The CO and continuum brightness temperatures imply that the nuclear gas disks have high area filling factors (~0.5-1) and extremely high visual extinctions (AVsimeq1000 mag). The molecular gas must be hot (≥40 K) and dense (≥104-5 cm-3), given the large mass and small scale height of the nuclear disks. The continuum data suggest that the large luminosity (be it starburst or active galactic nucleus) must originate within lesssim100 pc of the two nuclear gas disks, which were presumably formed through concentration of gas from the progenitor outer galaxy disks.