The masses of the supermassive black holes found in galaxy bulges are correlated with a multitude of galaxy properties(1,2), leading to suggestions that galaxies and black holes may evolve together(3). The number of reliably measured black-hole masses is small, and the number of methods for measuring them is limited(4), holding back attempts to understand this co-evolution. Directly measuring black-hole masses is currently possible with stellar kinematics (in early-type galaxies), ionized-gas kinematics (in some spiral and early-type galaxies(5-7)) and in rare objects that have central maser emission(8). Here we report that by modelling the effect of a black hole on the kinematics of molecular gas it is possible to fit interferometric observations of CO emission and thereby accurately estimate black-hole masses. We study the dynamics of the gas in the early-type galaxy NGC 4526, and obtain a best fit that requires the presence of a central dark object of 4.5(-3.1)(+4.2) x 10(8) solar masses, (3 sigma confidence limit). With the next-generation millimetre-wavelength interferometers these observations could be reproduced in galaxies out to 75 megaparsecs in less than 5 hours of observing time. The use of molecular gas as a kinematic tracer should thus allow one to estimate black-hole masses in hundreds of galaxies in the local Universe, many more than are accessible with current techniques.