  As part of the mm-Wave Interferometric Survey of Dark Object Masses (WISDOM)
project, we present an estimate of the mass of the supermassive black hole
(SMBH) in the nearby fast-rotating early-type galaxy NGC4697. This estimate is
based on Atacama Large Millimeter/submillimeter Array (ALMA) cycle-3
observations of the 12CO(2-1) emission line with a linear resolution of 29 pc
(0.53"). We find that NGC4697 hosts a small relaxed central molecular gas disc
with a mass of 1.6x10^7 Msun, co-spatial with the obscuring dust disc visible
in optical Hubble Space Telescope (HST) imaging. We also resolve thermal 1mm
continuum emission from the dust in this disc. NGC4697 is found to have a very
low molecular gas velocity dispersion, $\sigma_{gas}=1.65^{+0.68}_{-0.65}$
km/s. This seems to be partially because the giant molecular cloud mass
function is not fully sampled, but other mechanisms such as chemical
differentiation in a hard radiation field or morphological quenching also seem
to be required. We detect a Keplerian increase of the rotation of the molecular
gas in the very centre of NGC4697, and use forward modelling of the ALMA data
cube in a Bayesian framework with the KINematic Molecular Simulation (KinMS)
code to estimate a SMBH mass of ($1.3_{-0.17}^{+0.18})\times10^8$ Msun and an
i-band mass-to-light ratio of $2.14_{-0.05}^{+0.04}$ Msun/Lsun (at the 99%
confidence level). Our estimate of the SMBH mass is entirely consistent with
previous measurements from stellar kinematics. This increases confidence in the
growing number of SMBH mass estimates being obtained in the ALMA era.
