MANY galaxies are thought to contain massive black holes- exceeding ten million solar masses-at their centres1,2, but firm observational evidence has proved to be surprisingly elusive. The best evidence comes from observing gas or stars rotating rapidly within a small region around a central body. If the observed velocities are due solely to the gravitational force of the central body-as in the Solar System-then the mass of the central body can be readily calculated. Here we present observations of rotating gas near the centre of the galaxy NGC4258 (Ml06), which indicate the presence of a mass of 3.6 x 1077solar masses in a region less than 0.13 pc in radius. The volume-averaged mass density in this region exceeds by a factor of at least 40 that for any other black-hole candidate observed previously. These observations provide compelling evidence that a massive black hole exists at the centre of NGC4258.