The unique properties of water in the supercooled (metastable) state are not fully understood. In particular, the effects of solutes and mechanical pressure on the kinetics of the liquid-to-solid phase transition of supercooled water and aqueous solutions to ice have remained unresolved. Here we show from experimental data that the homogeneous nucleation of ice from supercooled aqueous solutions is independent of the nature of the solute, but depends only on the water activity of the solution--that is, the ratio between the water vapour pressures of the solution and of pure water under the same conditions. In addition, we show that the presence of solutes and the application of pressure have a very similar effect on ice nucleation. We present a thermodynamic theory for homogeneous ice nucleation, which expresses the nucleation rate coefficient as a function of water activity and pressure. Recent observations from clouds containing ice are in good agreement with our theory and our results should help to overcome one of the main weaknesses of numerical models of the atmosphere, the formulation of cloud processes.