Measurements of the high-energy cut-off in the coronal continuum of active galactic nuclei have long been elusive for all but a small number of the brightest examples. We present a direct measurement of the cut-off energy in the nuclear continuum of the nearby Seyfert 1.9 galaxy MCG-05-23-016 with unprecedented precision. The high sensitivity of NuSTAR up to 79 keV allows us to clearly disentangle the spectral curvature of the primary continuum from that of its reflection component. Using a simple phenomenological model for the hard X-ray spectrum, we constrain the cut-off energy to keV with 90% confidence. Testing for more complex models and nuisance parameters that could potentially influence the measurement, we find that the cut-off is detected robustly. We further use simple Comptonized plasma models to provide independent constraints for both the kinetic temperature of the electrons in the corona and its optical depth. At the 90% confidence level, we find kTe = 29 ± 2 keV and τe = 1.23 ± 0.08 assuming a slab (disk-like) geometry, and kTe = 25 ± 2 keV and τe = 3.5 ± 0.2 assuming a spherical geometry. Both geometries are found to fit the data equally well and their two principal physical parameters are correlated in both cases. With the optical depth in the τe ≳ 1 regime, the data are pushing the currently available theoretical models of the Comptonized plasma to the limits of their validity. Since the spectral features and variability arising from the inner accretion disk have been observed previously in MCG-05-23-016, the inferred high optical depth implies that a spherical or disk-like corona cannot be homogeneous.