Dense surface blooms of toxic cyanobacteria in eutrophic lakes may lead to mass mortalities of fish and birds, and provide a serious health threat for cattle, pets, and humans. It has been argued that global warming may increase the incidence of harmful algal blooms. Here, we report on a lake experiment where intermittent artificial mixing failed to control blooms of the harmful cyanobacterium Microcystis during the summer of 2003, one of the hottest summers ever recorded in Europe. To understand this failure, we develop a coupled biological-physical model investigating how competition for light between buoyant cyanobacteria, diatoms, and green algae in eutrophic lakes is affected by the meteorological conditions of this extreme summer heatwave. The model consists of a phytoplankton competition model coupled to a one-dimensional hydrodynamic model, driven by meteorological data. The model predicts that high temperatures favour cyanobacteria directly, through increased growth rates. Moreover, high temperatures also increase the stability of the water column, thereby reducing vertical turbulent mixing, which shifts the competitive balance in favour of buoyant cyanobacteria. Through these direct and indirect temperature effects, in combination with reduced wind speed and reduced cloudiness, summer heatwaves boost the development of harmful cyanobacterial blooms. These findings warn that climate change is likely to yield an increased threat of harmful cyanobacteria in eutrophic freshwater ecosystems.