A one-dimensional, quasi-steady, directional solidification problem is considered for the purpose of analyzing the relationship between the solution resulting from a phase-field model to that from a sharp-interface model whose exact solution is known. In the phase-field case, the quasi-steady approximation is coupled with a small-interface thickness boundary-layer expansion. By tuning the kinetic parameter, we develop phase-field temperature profiles that agree with the sharp-interface profiles, except near the solidification front, where there is smoothing over the diffuse interface and no jump in the temperature gradients. This results in agreement between the locations of the sharp-interface and phase-field solidification fronts.