An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Péclet number for small values of the surface-tension parameter. The velocity selection is caused by the effect of transcendentally small terms that are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small-Péclet-number analytical results of other investigators, although there are some discrepancies in details. It also addresses questions raised by a recent investigator on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.