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We report 18 cm VLBI continuum imaging observations of Arp 220, the prototype luminous infrared galaxy (log Lfir = 12.11 L☉). In previous work, we showed that Arp 220 has compact, high-Tb nuclear radio emission that might be interpreted as a dust-enshrouded active galactic nucleus (AGN) radio core, or, alternately, as multiple, very luminous radio supernovae from a very active nuclear starburst. In this work, we present a new 18 cm VLBI image, with 3×8 mas angular resolution, showing approximately a dozen unresolved sources, S18 cm = 0.2-1.2 mJy, within a 02 × 04 (75×150 pc) region centered on the NW nucleus of this merging system. At least two additional sources are detected in the SE nucleus. These point sources account for about 3% of the total 18 cm radio emission associated with Arp 220 and for all the estimated radio flux density with Tb > 106 K. No other 18 cm emission is detected on scales from 3 to 100 mas (1-30 pc). We interpret these compact radio sources as luminous radio supernovae of the class in which RSN 1986J is a prototype. This interpretation is consistent with a simple starburst model for the infrared luminosity of Arp 220 that has a star formation rate of 50-100 M☉ yr-1 and a luminous supernova rate, νsn = 1.75-3.5 yr-1. In this model prescription, virtually all supernova explosions in Arp 220 must result in luminous RSNe, comparable to the most luminous RSNe observed. We discuss possible mechanisms for the origin of very luminous RSNe in luminous infrared galaxies and suggest that it is likely due to the dense, compact starburst environment. Although our observations do not rule out the presence of an AGN that may contribute to the infrared luminosity in Arp 220, it is not necessary to appeal to AGN activity to account for the overall radio/infrared characteristics of Arp 220.