The centre of the Milky Way hosts several high-energy processes that have strongly affected the inner regions of our Galaxy. Activity from the super-massive black hole at the Galactic Centre, which is coincident with the radio source Sagittarius A*, and stellar feedback from the inner molecular ring(1) expel matter and energy from the disk in the form of a galactic wind(2). Multiphase gas has been observed within this outflow, including hot highly ionized(3,4) (temperatures of about 106 kelvin), warm ionized(5,6) (10(4) to 10(5) kelvin) and cool atomic(7,8) (10(3) to 10(4) kelvin) gas. However, so far there has been no evidence of the cold dense molecular phase (10 to 100 kelvin). Here we report observations of molecular gas outflowing from the centre of our Galaxy. This cold material is associated with atomic hydrogen clouds travelling in the nuclear wind(8). The morphology and the kinematics of the molecular gas, resolved on a scale of about one parsec, indicate that these clouds are mixing with the warmer medium and are possibly being disrupted. The data also suggest that the mass of the molecular gas outflow is not negligible and could affect the rate of star formation in the central regions of the Galaxy. The presence of this cold, dense and high-velocity gas is puzzling, because neither Sagittarius A* at its current level of activity nor star formation in the inner Galaxy seems to be a viable source for this material.