Frost flowers are clusters of ice crystals found on freshly formed sea ice and occasionally on frozen lakes. They belong to a class of vapour-related phenomena that includes freezing fog, hoar frost and dew. It has hitherto been supposed that they form by condensation from a supersaturated atmosphere or from water wicked up through porous sea ice. Here we show that they can form on solid, pure ice sublimating into an unsaturated atmosphere. We derive a general regime diagram showing the atmospheric conditions under which the different vapour-related phenomena occur and confirm our predictions of frost-flower formation with a series of laboratory experiments. Our results can be used in climate models to predict occurrence of frost flowers, which significantly enhance albedo and provide the substrate for chemical production of ozone-depleting bromine monoxide, and in paleo-climate reconstructions by relating observations of sea-salt aerosols in ice cores to atmospheric conditions.