The stagnation-flow Stefan solidification problem is defined and investigated. By applying the method of instantaneous similarity, the temperature field, the solid-liquid interface location and its growth rate, valid for the initial stages of solidification, are obtained. Furthermore, with the use of the quasi-steady approximation, a solution of the problem valid for the final stages of solidification is obtained. The analysis reveals a fundamental difference between the stagnation-flow solidification behavior and that in the classical Stefan solidification problem. Both methods of solution are used to show that the solidification front grows asymptotically to a finite maximum value as time goes to infinity. For large values of time, both methods yield the same temperature distribution and the same value of the solid phase thickness, which are independent of the Stefan number.