The paper presents a theory of the unsteady growth of spherical crystals in supercooled melts and supersaturated solutions. Using the methods of differential series, expansion of unknown functions in a small parameter, and the application of the integral Laplace-Carson transform, the first two corrections to the fundamental contribution to the law for the crystal growth rate were determined. It is shown that the corrections found substantially change the growth rates of spherical crystals in metastable liquids.