We present new interferometer observations of the 12CO(1-0) and 12CO(2-1) line emission of NGC 1068 with a resolution of 0.7''. The molecular gas in the inner 5'' is resolved into a ring with two bright knots east and west of the nuclear continuum emission. For the first time in NGC 1068, we can trace molecular gas at ~0.18'' (13pc) from the nucleus. The high velocities in this region imply an enclosed mass of ~10^8 M_solar. This value is consistent with a black hole mass of 1.7x10^7 M_solar plus a contribution from a compact nuclear stellar cluster. Perpendicular to the kinematic major axis optical images of NGC1068 show a bright, stellar, oval structure of eccentricity 0.8 and a deprojected length of 17kpc. Analysis of the rotation curve shows the CO spiral arms are at the inner Lindblad resonance of this bar-like structure. Inside the molecular spiral arms the CO kinematic axis changes direction probably in response to the 2.5kpc (deprojected) long stellar bar seen in the near infrared (NIR). The low velocity dispersion indicates the molecular gas is in a disk with a thickness of 10pc in the nuclear region and 100pc in the spiral arms. We constructed kinematic models for the molecular gas using elliptical orbits caused by a ~1'' nuclear bar and using tilted rings resulting in a warp. We find that the gas motions are consistent with either the warp or the bar models. However, because there is no evidence for a 1'' nuclear bar in NIR images, we favor the warp model. A warped CO disk can also explain the obscuration of the AGN, the extinction of light from the nuclear stellar cluster, and the observed NIR and mid-IR polarization. The model predicts the warped CO disk should become edge-on at a radius of 70pc, thereby creating a cavity for the ionization cone.