We estimate the mass of the intermediate-mass black hole at the heart of the dwarf elliptical galaxy NGC 404 using Atacama Large Millimetre/submillimetre Array (ALMA) observations of the molecular interstellar medium at an unprecedented linear resolution of ≈0.5 pc, in combination with existing stellar kinematic information. These ALMA observations reveal a central disc/torus of molecular gas clearly rotating around the black hole. This disc is surrounded by a morphologically and kinematically complex flocculent distribution of molecular clouds, that we resolve in detail. Continuum emission is detected from the central parts of NGC 404, likely arising from the Rayleigh-Jeans tail of emission from dust around the nucleus, and potentially from dusty massive star-forming clumps at discrete locations in the disc. Several dynamical measurements of the black hole mass in this system have been made in the past, but they do not agree. We show here that both the observed molecular gas and stellar kinematics independently require a 5 × 105M black hole once we include the contribution of the molecular gas to the potential. Our best estimate comes from the high-resolution molecular gas kinematics, suggesting the black hole mass of this system is 5.5+-3.8 × 105M (at the 99 per cent confidence level), in good agreement with our revised stellar kinematic measurement and broadly consistent with extrapolations from the black hole mass-velocity dispersion and black hole mass-bulge mass relations. This highlights the need to accurately determine the mass and distribution of each dynamically important component around intermediate-mass black holes when attempting to estimate their masses.