Ultra-compact dwarf galaxies are among the densest stellar systems in the Universe. These systems have masses of up to 2 × 108 solar masses, but half-light radii of just 3-50 parsecs. Dynamical mass estimates show that many such dwarfs are more massive than expected from their luminosity. It remains unclear whether these high dynamical mass estimates arise because of the presence of supermassive black holes or result from a non-standard stellar initial mass function that causes the average stellar mass to be higher than expected. Here we report adaptive optics kinematic data of the ultra-compact dwarf galaxy M60-UCD1 that show a central velocity dispersion peak exceeding 100 kilometres per second and modest rotation. Dynamical modelling of these data reveals the presence of a supermassive black hole with a mass of 2.1 × 107 solar masses. This is 15 per cent of the object's total mass. The high black hole mass and mass fraction suggest that M60-UCD1 is the stripped nucleus of a galaxy. Our analysis also shows that M60-UCD1's stellar mass is consistent with its luminosity, implying a large population of previously unrecognized supermassive black holes in other ultra-compact dwarf galaxies.