Dwarf satellite galaxies are thought to be the remnants of the population of primordial structures that coalesced to form giant galaxies like the Milky Way. It has previously been suspected that dwarf galaxies may not be isotropically distributed around our Galaxy, because several are correlated with streams of H I emission, and may form coplanar groups. These suspicions are supported by recent analyses. It has been claimed that the apparently planar distribution of satellites is not predicted within standard cosmology, and cannot simply represent a memory of past coherent accretion. However, other studies dispute this conclusion. Here we report the existence of a planar subgroup of satellites in the Andromeda galaxy (M 31), comprising about half of the population. The structure is at least 400 kiloparsecs in diameter, but also extremely thin, with a perpendicular scatter of less than 14.1 kiloparsecs. Radial velocity measurements reveal that the satellites in this structure have the same sense of rotation about their host. This shows conclusively that substantial numbers of dwarf satellite galaxies share the same dynamical orbital properties and direction of angular momentum. Intriguingly, the plane we identify is approximately aligned with the pole of the Milky Way's disk and with the vector between the Milky Way and Andromeda.