Hydrogen radicals are produced in the martian atmosphere by the photolysis of water vapour and subsequently initiate catalytic cycles that recycle carbon dioxide from its photolysis product carbon monoxide. These processes provide a qualitative explanation for the stability of the atmosphere of Mars, which contains 95 per cent carbon dioxide. Balancing carbon dioxide production and loss based on our current understanding of the gas-phase chemistry in the martian atmosphere has, however, proven to be difficult. Interactions between gaseous chemical species and ice cloud particles have been shown to be key factors in the loss of polar ozone observed in the Earth's stratosphere, and may significantly perturb the chemistry of the Earth's upper troposphere. Water-ice clouds are also commonly observed in the atmosphere of Mars and it has been suggested previously that heterogeneous chemistry could have an important impact on the composition of the martian atmosphere. Here we use a state-of-the-art general circulation model together with new observations of the martian ozone layer to show that model simulations that include chemical reactions occurring on ice clouds lead to much improved quantitative agreement with observed martian ozone levels in comparison with model simulations based on gas-phase chemistry alone. Ozone is readily destroyed by hydrogen radicals and is therefore a sensitive tracer of the chemistry that regulates the atmosphere of Mars. Our results suggest that heterogeneous chemistry on ice clouds plays an important role in controlling the stability and composition of the martian atmosphere.