We present high angular resolution (0.7") observations made with the Very Large Array (VLA) of the radio recombination line (RRL) H53alpha and radio continuum emission at 43 GHz from the ultraluminous infrared galaxy (ULIRG) Arp 220. The 43 GHz continuum emission shows a compact structure (~2") with two peaks separated by ~1", the east (E) and west (W) components, that correspond to each galactic nucleus of the merger. The spectral indices for both the E and W components, using radio continuum images at 8.3 and 43 GHz are typical of synchrotron emission (alpha~-1.0). Our 43 GHz continuum and H53alpha line observations confirm the flux densities predicted by the models proposed by Anantharamaiah et al. (2000). This agreement with the models implies the presence of high-density (~10⁵ cm^-3) compact H II regions (~0.1 pc) in Arp 220. The integrated H53alpha line emission is stronger toward the nonthermal radio continuum peaks, which are also coincident with the peaks of molecular emission of the H₂CO. The coincidence between the integrated H53alpha and the H₂CO maser line emission suggests that the recent star-forming regions, traced by the high-density gas, are located mainly in regions that are close to the two radio continuum peaks. A velocity gradient of ~0.30 km s^-1 pc^-1 in the H53alpha RRL is observed toward the E component, and a second velocity gradient of ~0.15 km s^-1 pc^-1 is detected toward the W component. The orientations of these velocity gradients are in agreement with previous CO, H I, and OH observations. The kinematics of the high-density ionized gas traced by the H53alpha line are consistent with two counterrotating disks as suggested by the CO and H I observations.