Residing in the centre of the Milky Way, Sagittarius A* (Sgr A*) is the closest massive black hole1 (MBH). Its vicinity has allowed measuring individual stellar orbits around it2–4. The stars act as test particles and probe the gravitational potential around the 4.3 × 106M⊙ MBH. These observations have determined the central mass to sub-per-cent precision5, and the mildly relativistic motions of stars have given access to the dominant relativistic corrections, the gravitational redshift6,7, the transverse Doppler effect and the prograde precession imposed by the Schwarzschild metric nature of the potential8. These effects are of order β2 = (v/c)2 (for velocity v and speed of light c). The Kerr metric for a rotating black hole leads to corrections of order β3. Here, we report the discovery of a faint main-sequence star (mK = 19.3), S301, on an 8.7-year orbit and with small enough a pericentre distance, such that the peak velocity of the star reaches 25,000 km s−1. Within the measurement abilities of current near-infrared interferometry and future spectroscopy on an extremely large telescope, the motion of S301 is directly sensitive to the spin of Sgr A*. The high eccentricity of S301 suggests that it is the captured component of a binary that was torn apart by the Hills mechanism. The discovery of the fast-orbiting star S301 near Sagittarius A* offers a promising new probe of the central black hole spin of the Milky Way through relativistic orbit measurements.