This paper presents new CO interferometer data revealing that molecular gas in ultraluminous infrared (IR) galaxies is distributed in rotating nuclear disks or rings. The data provide disk radii, kinematic major axes, rotation speeds, enclosed dynamical masses, and gas masses. The CO lines are subthermally excited and moderately opaque (τ = 4 to 10), indicating the gas is in a warm, intercloud medium rather than self-gravitating clouds. Gas masses are derived from radiative transfer models, yielding values about five times lower than standard methods. The ratio of gas to dynamical mass is approximately 1/6, and the maximum ratio of gas to total mass surface density is 1/3. The galaxies VII Zw 31, Arp 193, and IRAS 10565+2448 show evidence for rotating molecular rings with a central gap. An extreme starburst class is identified, characterized by a size of 100 pc, gas mass of ~10⁹ M☉, and IR luminosity of ~3×10¹¹ L☉ from OB stars. Four extreme starbursts are found in the three closest galaxies in the sample, including Arp 220, Arp 193, and Mrk 273. These are the most prodigious star formation events in the local universe, each representing about 1000 times as many OB stars as 30 Doradus. In Mrk 231, the CO(2–1) velocity diagram shows a 1"·2-diameter inner disk and a 3"-diameter outer disk. The narrow CO linewidth, single-peak line profile, and velocity gradients indicate the molecular disk is nearly face-on, yielding low optical and UV extinction to the AGN. The FIR luminosity of Mrk 231 is powered by a starburst, not the AGN. In Mrk 273, the CO(1–0) maps show long streamers of radius 5 kpc (7") with velocity gradients north-south, and a nuclear disk of radius 400 pc (0.6") with velocity gradients east-west. The nuclear disk contains a bright CO core of radius 120 pc (0.2"). In Arp 220, the CO and 1.3 mm continuum maps show two "nuclei" embedded in a central ring or disk at p.a. 50° and a fainter structure extending 7" (3 kpc) to the east. Models of the CO and dust flux indicate the two K-band sources contain high-density gas with n(H₂) = 2×10⁴ cm⁻³. These sources are more likely to be compact, extreme starburst regions containing 10⁹ M☉This paper presents new CO interferometer data revealing that molecular gas in ultraluminous infrared (IR) galaxies is distributed in rotating nuclear disks or rings. The data provide disk radii, kinematic major axes, rotation speeds, enclosed dynamical masses, and gas masses. The CO lines are subthermally excited and moderately opaque (τ = 4 to 10), indicating the gas is in a warm, intercloud medium rather than self-gravitating clouds. Gas masses are derived from radiative transfer models, yielding values about five times lower than standard methods. The ratio of gas to dynamical mass is approximately 1/6, and the maximum ratio of gas to total mass surface density is 1/3. The galaxies VII Zw 31, Arp 193, and IRAS 10565+2448 show evidence for rotating molecular rings with a central gap. An extreme starburst class is identified, characterized by a size of 100 pc, gas mass of ~10⁹ M☉, and IR luminosity of ~3×10¹¹ L☉ from OB stars. Four extreme starbursts are found in the three closest galaxies in the sample, including Arp 220, Arp 193, and Mrk 273. These are the most prodigious star formation events in the local universe, each representing about 1000 times as many OB stars as 30 Doradus. In Mrk 231, the CO(2–1) velocity diagram shows a 1"·2-diameter inner disk and a 3"-diameter outer disk. The narrow CO linewidth, single-peak line profile, and velocity gradients indicate the molecular disk is nearly face-on, yielding low optical and UV extinction to the AGN. The FIR luminosity of Mrk 231 is powered by a starburst, not the AGN. In Mrk 273, the CO(1–0) maps show long streamers of radius 5 kpc (7") with velocity gradients north-south, and a nuclear disk of radius 400 pc (0.6") with velocity gradients east-west. The nuclear disk contains a bright CO core of radius 120 pc (0.2"). In Arp 220, the CO and 1.3 mm continuum maps show two "nuclei" embedded in a central ring or disk at p.a. 50° and a fainter structure extending 7" (3 kpc) to the east. Models of the CO and dust flux indicate the two K-band sources contain high-density gas with n(H₂) = 2×10⁴ cm⁻³. These sources are more likely to be compact, extreme starburst regions containing 10⁹ M☉