  We estimate the mass of the intermediate-mass black hole at the heart of the
dwarf elliptical galaxy NGC 404 using Atacama Large Millimeter/submillimeter
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 ~5x10$^5$ Msun
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$^{+4.1}_{-3.8}\times$10$^5$ Msun (at the 99% 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.
