In the steel casting process, the deficient surface temperature history of the solidifying steel slab can induce the transverse surface cracks, while the improper relative location of the solidification front inside the slab down the caster can create the internal cracks. Furthermore, if the slab is not fully solidified when it leaves the support rolls of the caster, the pressure from the molten steel will cause the steel shell to bulge out drastically, creating a defect called "whale", which damages the casting machine and causes a long work stoppage. Therefore, regulation of both the steel temperature and the liquid-solid interface location history is the key to maintaining the high steel quality and operational safety. This regulation can be achieved by the water spray cooling. The latter, however, is characterized by saturation when the water flow rate reaches its available maximum. The previous work of the authors started addressing this problem by presenting the full state feedback enthalpy-based control law for the two-sided Stefan problem with actuator saturation, describing the steel slab solidification under spray rate constraint. However, in the actual system, the full state feedback is not possible, since only the solid boundary temperature sensing is available. The present work closes this fundamental gap by combining a full state controller with an observer based on the temperature of the solid boundary. This combination produces the output feedback control law capable of tracking the desired temperature and solidification front trajectory under input saturation in the two-sided Stefan problem. The closed-loop convergence of the temperature and the interface errors for the output feedback system obtained are proven. Simulation shows the exponential-like trajectory convergence attained by the implementable smooth bounded control signals.