Proxima and α Centauri AB have almost identical distances and proper motions with respect to the Sun. Although the probability of such similar parameters is, in principle, very low, the question as to whether they actually form a single gravitationally bound triple system has been open since the discovery of Proxima one century ago. Owing to HARPS high-precision absolute radial velocity measurements and the recent revision of the parameters of the α Cen pair, we show that Proxima and α Cen are gravitationally bound with a high degree of confidence. The orbital period of Proxima is ≈ 550 000 years. With an eccentricity of 0.50 +0.08 −0.09 , Proxima comes within 4.3 +1.1 −0.9 kau of α Cen at periastron, and is currently close to apastron (13.0 +0.3 −0.1 kau). This orbital motion may have influenced the formation or evolution of the recently discovered planet orbiting Proxima, as well as circumbinary planet formation around α Cen.