We resolve the extended X-ray emission from the prototypical ultraluminous infrared galaxy Arp 220. Extended, faint edge-brightened, soft X-ray lobes outside the optical galaxy are observed to a distance of 10 to 15 kpc on each side of the nuclear region. Bright plumes inside the optical isophotes coincide with the optical line emission and extend 11 kpc from end to end across the nucleus. The data for the plumes cannot be fit by a single temperature plasma, and display a range of temperatures from 0.2 to 1 keV. The plumes emerge from bright, diffuse circumnuclear emission in the inner 3 kpc centered on the Halpha peak, which is displaced from the radio nuclei. There is a close morphological correspondence between the Halpha and soft X-ray emission on all spatial scales. We interpret the plumes as a starburst-driven superwind, and discuss two interpretations of the emission from the lobes in the context of simulations of the merger dynamics of Arp 220.