ABSTRACT As part of the Measuring Black Holes in below Milky Way-mass (M⋆) galaxies (MBHBM⋆) Project, we present a dynamical measurement of the supermassive black hole (SMBH) mass in the nearby lenticular galaxy NGC 3593, using cold molecular gas 12CO(2-1) emission observed at an angular resolution of ≈0${_{.}^{\prime\prime}}$3 (≈10 pc) with the Atacama Large Millimeter/submillimeter Array (ALMA). Our ALMA observations reveal a circumnuclear molecular gas disc (CND) elongated along the galaxy major axis and rotating around the SMBH. This CND has a relatively low-velocity dispersion (≲10 km s−1) and is morphologically complex, with clumps having higher integrated intensities and velocity dispersions (≲25 km s−1). These clumps are distributed along the ridges of a two-arm/bi-symmetric spiral pattern surrounded by a larger ring-like structure (radius r ≈ 10 arcsec or ≈350 pc). This pattern likely plays an important role to bridge the molecular gas reservoirs in the CND and beyond (10 ≲ r ≲ 35 arcsec or 350 pc ≲ r ≲ 1.2 kpc). Using dynamical modelling, the molecular gas kinematics allow us to infer an SMBH mass $M_{\rm BH}=2.40_{-1.05}^{+1.87}\times 10^6$ M⊙ (only statistical uncertainties at the 3σ level). We also detect a massive core of cold molecular gas (CMC) of mass MCMC = (5.4 ± 1.2) × 106 M⊙ and effective (half-mass) radius rCMC,e = 11.2 ± 2.8 pc, co-spatial with a nuclear star cluster (NSC) of mass MNSC = (1.67 ± 0.48) × 107 M⊙ and effective radius rNSC,e = 5.0 ± 1.0 pc (or 0${_{.}^{\prime\prime}}$15 ± 0${_{.}^{\prime\prime}}$03). The mass profiles of the CMC and NSC are well described by Sérsic functions with indices 1−1.4. Our MBH and MNSC estimates for NGC 3593 agree well with the recently compiled MBH–MNSC scaling relation. Although the MNSC uncertainty is twice the inferred MBH, the rapid central rise of the rotation velocities of the CND (as the radius decreases) clearly suggests an SMBH. Indeed, our dynamical models show that even if MNSC is at the upper end of its allowed range, the evidence for a BH does not vanish, but remains with a lower limit of MBH > 3 × 105 M⊙.