The phase transition between solid and liquid in an undercooled liquid leads to dendritic growth of the solid phase. The problem is modelled by the Stefan problem with a modified Gibbs-Thomson law, which includes the anisotropic mean curvature corresponding to a surface energy that depends on the direction of the interface normal. A finite element method for discretization of the Stefan problem is described which is based on a weak formulation of the anisotropic mean curvature flow. Numerical experiments with a nearly crystalline anisotropy are presented.