We have mapped the central region of NGC 4945 in the J = 2 → 1 transition of 12CO, 13CO, and C18O, as well as the continuum at ~1.3 mm, at an angular resolution of 5'' × 3'' with the Submillimeter Array. The relative proximity of NGC 4945 (distance of only 3.8 Mpc) permits a detailed study of the circumnuclear molecular gas and dust in a galaxy exhibiting both an active galactic nucleus (AGN; classified as Seyfert 2) and a circumnuclear starburst in an inclined ring with radius ~2.5'' (~50 pc). We infer the systemic velocity ~585 km s-1 from chanel maps and PV-diagrams. We find that all three molecular lines trace an inclined rotating disk with the major axis aligned with that of the starburst ring and large-scale galactic disk and which exhibits solid-body rotation within a radius of ~5'' (~95 pc). The rotation curve flattens beyond this radius, and the isovelocity contours exhibit an S-shaped asymmetry suggestive of a highly inclined bar, as has been invoked to produce a similar asymmetry observed on larger scales. We infer an inclination for the nuclear disk of 62° ± 2°, somewhat smaller than the inclination of the large-scale galactic disk of ~78°. The continuum emission at 1.3 mm also extends beyond the starburst ring and is dominated by thermal emission from dust. If it traces the same dust emitting in the far-infrared, then the bulk of this dust must be heated by star-formation activity rather than the AGN. We discover a kinematically decoupled component at the center of the disk with a radius smaller than 1.4'' (27 pc), but which spans approximately the same range of velocities as the surrounding disk. This component has a higher density than its surroundings and is a promising candidate for the circumnuclear molecular torus invoked by AGN unification models.