We present ALMA Band 6 12CO(2–1) line and rest-frame 232 GHz continuum observations of the nearby Compton-thick Seyfert galaxy NGC 5643 with angular resolutions 0.″11–0.″26 (9–21 pc). The CO(2–1) integrated line map reveals emission from the nuclear and circumnuclear region with a two-arm nuclear spiral extending ∼10″ on each side. The circumnuclear CO(2–1) kinematics can be fitted with a rotating disk, although there are regions with large residual velocities and/or velocity dispersions. The CO(2–1) line profiles of these regions show two different velocity components. One is ascribed to the circular component and the other to the interaction of the AGN outflow, as traced by the [O III]λ5007 Å emission, with molecular gas in the disk a few hundred parsecs from the AGN. On nuclear scales, we detected an inclined CO(2–1) disk (diameter 26 pc, FWHM) oriented almost in a north–south direction. The CO(2–1) nuclear kinematics can be fitted with a rotating disk that appears to be tilted with respect to the large-scale disk. There are strong non-circular motions in the central 0.″2–0.″3 with velocities of up to 110 km s‑1. In the absence of a nuclear bar, these motions could be explained as radial outflows in the nuclear disk. We estimate a total molecular gas mass for the nuclear disk of M(H2) = 1.1 × 107 M ⊙ and an H2 column density toward the location of the AGN of N(H2) ∼ 5 × 1023 cm‑2, for a standard CO-to-H2 conversion factor. We interpret this nuclear molecular gas disk as the obscuring torus of NGC 5643 as well as the collimating structure of the ionization cone.