We consider the directional solidification, in two dimensions, of a dilute binary alloy having a large anisotropy of surface energy, (i.e., orientations with negative surface stiffness), where the sample is pulled in the highest-energy direction. Linear stability analysis reveals that the planar state is thermodynamically prohibited, leading to a search for faceted solutions. Below the critical pulling speed associated with constitutional supercooling, a small-wavelength assumption allows the reduction of interface dynamics to a single PDE. Matched asymptotic analysis then reveals a family of faceted interface profiles, while variational arguments confirm a small optimal wavelength. Questions on dynamic behavior lead to the derivation of a gradient-descent dynamics and an associated facet-velocity law. This reveals that faceted steady solutions are stable in the absence of supercooling, while coarsening replaces cell formation as the instability mechanism when supercooling is reached. (C) 2007 Elsevier B.V. All rights reserved.