High-resolution CO(1 → 0) observations of five ultraluminous infrared galaxies [ULIGs; L_IR 10¹² L] with double nuclei are analyzed. These sources constitute a complete subset of local ULIGs expected to be in an intermediate stage of merging and are selected to have projected nuclear separations of 20-54 (3-5 kpc) so they could be resolved with the Owens Valley Radio Observatory Millimeter Array. The observed pairs include two mergers with cool far-infrared colors (25-60 μm flux density ratio f_{25 μm}/f_{60 μm} < 0.2) from the Infrared Astronomical Satellite (IRAS) Bright Galaxy Sample (IRAS 12112+0305 and IRAS 14348-1447) and three mergers with warm infrared dust temperatures (f_{25 μm}/f_{60 μm} 0.2) selected from the IRAS Warm Galaxy Sample (IRAS 08572+3915, IRAS 13451+1232 = PKS 1345+12, and IRAS 13536+1836 = Mrk 463). These ULIGs are further distinguished by the presence of pairs of active nuclei; among the 10 nuclei, nine have Seyfert or LINER classifications and one is unclassified. Molecular gas is detected only on the redder, more radio-luminous nucleus of the warm objects, whereas both nuclei of the cool ULIGs are detected in CO. The inferred molecular gas masses for the detected nuclei are × 10¹⁰ M, and the undetected nuclei have molecular gas masses at least 1.2-2.8 times less than that of their CO-luminous companions. Upper limits on the extent of the CO-emitting regions of each detected nucleus range from 2 to 4 kpc, which is about 3-6 times smaller than the average effective CO diameter of nearby spiral galaxies. This is strong evidence that the high concentration of molecular gas is the result of tidal dissipation in ongoing mergers. There is no correlation between the optical emission-line classification of the nuclei (i.e., Seyfert, LINER, or H II) and the presence of detectable molecular gas; however, there is a clear indication that the relative amount of molecular gas increases with the relative level of activity as measured via radio power and optical/near-infrared emission-line strength. Star formation rates are estimated to be in the range ~30-290 M yr^-1 nucleus^-1 by making an assumption that the radio and infrared emission arise from supernovae and dust heating by massive stars, respectively; corresponding gas consumption timescales are × 10⁷ yr. The nuclei detected in CO are extremely red at near-infrared wavelengths, suggestive of much dustier environments than in the companions undetected in CO. Column density estimates are N ~ 10²⁴-10²⁵ cm^-2, which correspond to more than 1000 magnitudes of extinction toward the nuclei at visual wavelengths. Finally, the molecular gas mass densities and line-of-sight velocity dispersions show significant overlap with stellar densities and line-of-sight stellar velocity dispersions of local elliptical galaxies with M_V < -19 mag, including rapidly rotating ellipticals with disky isophotes and power-law light profiles, as well as slowly rotating ellipticals with boxy isophotes and cores. This provides strong evidence that the CO-rich nuclei of these ULIGs have the phase-space density of gas necessary to form the stellar cores of elliptical galaxies.