The centre of the Milky Way is the site of several high-energy processes that have strongly impacted the inner regions of our Galaxy. Activity from the super-massive black hole, Sgr A*, and/or stellar feedback from the inner molecular ring expel matter and energy from the disc in the form of a galactic wind. Multiphase gas has been observed within this outflow, from hot highly-ionized, to warm ionized and cool atomic gas. To date, however, there has been no evidence of the cold and dense molecular phase. Here we report the first detection of molecular gas outflowing from the centre of our Galaxy. This cold material is associated with atomic hydrogen clouds travelling in the nuclear wind. The morphology and the kinematics of the molecular gas, resolved on ~1 pc scale, indicate that these clouds are mixing with the warmer medium and are possibly being disrupted. The data also suggest that the mass of molecular gas driven out is not negligible and could impact the rate of star formation in the central regions. The presence of this cold, dense, high-velocity gas is puzzling, as neither Sgr A* at its current level of activity, nor star formation in the inner Galaxy seem viable sources for this material.