The centre of the Milky Way Galaxy hosts a black hole with a solar mass of about 4 million (Sagittarius A* (Sgr A)) that is very quiescent at present with a luminosity many orders of magnitude below those of active galactic nuclei1. Reflection of X-rays from Sgr A* by dense gas in the Galactic Centre region offers a means to study its past flaring activity on timescales of hundreds and thousands of years2. The shape of the X-ray continuum and the strong fluorescent iron line observed from giant molecular clouds in the vicinity of Sgr A* are consistent with the reflection scenario3-5. If this interpretation is correct, the reflected continuum emission should be polarized6. Here we report observations of polarized X-ray emission in the direct...