Context. The central engines of many active galactic nuclei (AGNs) are thought to be surrounded by warm molecular and dusty material in an axisymmetric geometry, thus explaining part of the observational diversity of active nuclei. Aims. We aim to shed light on the physical properties and kinematics of the molecular material in the nucleus of one of the closest type 2 active galaxies. Methods. To this end, we obtained high angular resolution Atacama Large Millimeter/submillimeter Array (ALMA) observations of the nucleus of the Circinus galaxy. The observations map the emission at 350 GHz and 690 GHz with spatial resolutions of ∼3.8 pc and ∼2.2 pc, respectively. Results. The continuum emission traces cold (T ≲ 100 K) dust in a circumnuclear disk with spiral arms on scales of 25 pc, plus a marginally resolved nuclear emission peak. The latter is not extended in polar direction as claimed based on earlier ALMA observations. A significant amount (of the order of 40%) of the 350 GHz emission is not related to dust, but most likely free-free emission instead. We detect CO(3−2) and CO(6−5) as well as HCO⁺(4−3), HCN(4−3), and CS(4−3). The CO emission is extended, showing a spiral pattern, similar to the extended dust emission. Towards the nucleus, CO is excited to higher transitions and its emission is self-absorbed, leading to an apparent hole in the CO(3−2) but not the CO(6−5) emission. On the other hand, the high gas density tracers HCO⁺, HCN, and CS show a strong, yet unresolved (≲4 pc) concentration of the emission at the nucleus, pointing at a very small ‘torus’. The kinematics are dominated by rotation and point at a geometrically thin disk down to the resolution limit of our observations. In contrast to several other AGNs, no HCN enhancement is found towards the nucleus. Conclusions. The Circinus nucleus is therefore composed of at least two distinct components: (1) an optically thin, warm outflow of ionised gas containing clouds of dust which are responsible for the polar mid-infrared emission, but not seen at submillimetre wavelengths; and (2) a cold molecular and dusty disk, traced by submillimetre continuum and line emission. The latter is responsible for the bulk of the obscuration of the nucleus. These findings support the most recent radiative transfer calculations of the obscuring structures in AGNs, which find a similar two-component structure.