We determine the stability of a nonequilibrium interface between two coexisting solid phases in the presence of a weak external field. Starting at the coarse-grained (Cahn-Hilliard) level, we use the method of matched asymptotics to derive the macroscopic interracial dynamics. We then show that the external field leads to an instability due to flux along the interface, in contrast with the more common Mullins Sekerka type instability, which involves fluxes normal to the interface. We also find that the external field produces an important modification of the Gibbs-Thomson relation. With these results, we perform the linear stability analysis for an approximately flat interface. If the field is tangent to the interface, the modification of the Gibbs-Thomson relation is important and the interface is stabilized. If the field is normal to the interface, the surface flux is important, and the effect can be stabilizing or destabilizing, but the orion-rational dependence is opposite what would be obtained if the Mullins-Sekerka instability dominates. Numerical simulations are performed to study the effect of the surface current and are in agreement with our analytical results.