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New CO interferometer data show that the molecular gas in infrared ultraluminous galaxies is in rotating nuclear disks or rings. The CO maps yield disk radii, kinematic major axes, rotation speeds, enclosed dynamical masses, and gas masses. The CO brightness temperatures, the double-peaked CO line proles, the limits on thermal continuum ux from dust, and the constraint that the gas mass must be less than the dynamical mass all indicate that the CO lines are subthermally excited and moderately opaque (q \ 4 to 10). We t kinematic models in which most of the CO ux comes from a moderatedensity warm intercloud medium, rather than from self-gravitating clouds. Typical ring radii are 300 to 800 pc. We derive gas masses not from a standard CO-to-mass ratio, but from a model of radiative transfer through subthermally excited CO in the molecular disks. This model yields gas masses of D5 ] 109 D5 times lower than the standard method, and a ratio (K km s~1 M _ , M gas /L CO @ B0.8 M _ pc2)~1. In the nuclear disks, we derive a ratio of gas to dynamical mass of and a M gas /M dyn B 1/6, maximum ratio of gas to total mass surface density, of 1/3. For the galaxies VII Zw 31, Arp 193, k/k tot , and IRAS 10565]2448, the CO position-velocity diagrams provide good evidence for rotating molecular rings with a central gap. In addition to the rotating central rings or disks, a new class of star formation region is identied, which we call an extreme starburst. These have a characteristic sizes of only 100 pc, with about 109 of gas and an IR luminosity of B3 ] 1011 from recently formed OB stars. Four M _ L _ extreme starbursts are identied in the 3 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 (21) velocity diagram along the line of nodes shows a diameter inner disk and a 3A diameter outer disk. The narrow CO line 1A .2 width, the single-peak line prole, the equality of the major and minor axes, and the observed velocity gradients all imply that the molecular disk is nearly face-on, yielding low optical and UV extinction to the active galactic nucleus (AGN). Such a geometry means that the molecular disk cannot be heated by the AGN ; the far-infrared (FIR) luminosity of Mrk 231 is powered by a starburst, not the AGN. In Mrk 273, the CO (10) maps show long streamers of radius 5 kpc (7A) with velocity gradients north-south, and a nuclear disk of radius 400 pc with velocity gradients east-west. The nuclear disk contains a (0A .6) bright CO core of radius 120 pc