A new formulation is presented of the thermodynamical theory of heterogeneous ice crystal nucleation in clouds by freezing. This theory unifies and explains the empirical ice nuclei dependence on temperature and supersaturation, predicts crystal formation via condensation-freezing at a subsaturation over water. The theory also explains observations of high nucleation rates and crystal concentrations at warm (-5>-12 ¿C) temperatures when the splintering mechanism may be not effective. This theory can be applied to parameterizations for use in cloud and climate models. ¿ 2000 American Geophysical Union