- We present Atacama Large Millimeter Array (ALMA) Cycle-0 observations of the CO J = 6-5 line in the advanced galaxy merger Arp 220. This line traces warm molecular gas, which dominates the total CO luminosity. The CO emission from the two nuclei is well resolved by the 0.39" x 0.22" beam and the exceptional sensitivity and spatial/spectral resolution reveal new complex features in the morphology and kinematics of the warm gas. The line profiles are asymmetric between the red and blue sides of the nuclear disks and the peak of the line emission is offset from the peak of the continuum emission in both nuclei by about 100 pc in the same direction. CO self-absorption is detected at the centers of both nuclei but it is much deeper in the eastern nucleus. We also clearly detect strong, highly redshifted CO absorption located near the southwest side of each nucleus. For the eastern nucleus, we reproduce the major line profile features with a simple kinematic model of a highly turbulent, rotating disk with a substantial line center optical depth and a large gradient in the excitation temperature. The red/blue asymmetries and line-to-continuum offset are likely produced by absorption of the blue (SW) sides of the two nuclei by blue-shifted, foreground molecular gas; the mass of the absorber is comparable to the nuclear warm gas mass (10^8 M_solar). We measure an unusually high L_CO/L_FIR ratio in the eastern nucleus, suggesting there is an additional energy source, such as mechanical energy from shocks, present in this nucleus.