Clusters of galaxies are the most massive gravitationally bound objects in the Universe and are still forming. They are thus important probes(1) of cosmological parameters and many astrophysical processes. However, knowledge of the dynamics of the pervasive hot gas, the mass of which is much larger than the combined mass of all the stars in the cluster, is lacking. Such knowledge would enable insights into the injection of mechanical energy by the central supermassive black hole and the use of hydrostatic equilibrium for determining cluster masses. X-rays from the core of the Perseus cluster are emitted by the 50-million-kelvin diffuse hot plasma filling its gravitational potential well. The active galactic nucleus of the central galaxy NGC 1275 is pumping jetted energy into the surrounding intracluster medium, creating buoyant bubbles filled with relativistic plasma. These bubbles probably induce motions in the intracluster medium and heat the inner gas, preventing runaway radiative cooling-a process known as active galactic nucleus feedback(2-6). Here we report X-ray observations of the core of the Perseus cluster, which reveal a remarkably quiescent atmosphere in which the gas has a line-of-sight velocity dispersion of 164 +/- 10 kilometres per second in the region 30-60 kiloparsecs from the central nucleus. A gradient in the line-of-sight velocity of 150 +/- 70 kilometres per second is found across the 60-kiloparsec image of the cluster core. Turbulent pressure support in the gas is four per cent of the thermodynamic pressure, with large-scale shear at most doubling this estimate. We infer that a total cluster mass determined from hydrostatic equilibrium in a central region would require little correction for turbulent pressure.