We use a fast solver for the heat equation in free-space to compute the thermal field in a simulation of crystal growth at low undercoolings. The solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient, free space heat equation with a smooth source term. The computational cost and memory requirements of the new solver are reasonable compared to standard methods and allow one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field. This can result in significant computational savings in situations where the crystals grow slowly relative to the expansion of the thermal field and may make parameter regimes typically encountered in experimental situations more accessible to numerical treatment. We demonstrate the method on the problem of anisotropic, dendritic growth at low undercooling in two dimensions.