We present a finite element approximation for a continuum model of snow crystalgrowth. The model is given by a one-sided Stefanproblem with a fully anisotropic Gibbs–Thomson law and kinetic undercooling.Our approximation, which couples a parametric approximation of the movingboundary with a finite element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys verygood mesh properties which means that no mesh smoothing is necessary inpractice. In our numerical computations we simulate snow crystal growth in two and threespace dimensions. On choosing realistic physical parameters, we are able toproduce several distinctive types of snow crystal morphologies.