  In this paper, a backstepping observer and an output feedback control law are
designed for the stabilization of the one-phase Stefan problem. The present
result is an improvement of the recent full state feedback backstepping
controller proposed in our previous contribution. The one-phase Stefan problem
describes the time-evolution of a temperature profile in a liquid-solid
material and its liquid-solid moving interface. This phase transition problem
is mathematically formulated as a 1-D diffusion Partial Differential Equation
(PDE) of the melting zone defined on a time-varying spatial domain described by
an Ordinary Differential Equation (ODE). We propose a backstepping observer
allowing to estimate the temperature profile along the melting zone based on
the available measurement, namely, the solid phase length. The designed
observer and the output feedback controller ensure the exponential stability of
the estimation errors, the moving interface, and the ${\cal H}_1$-norm of the
distributed temperature while keeping physical constraints, which is shown with
the restriction on the gain parameter of the observer and the setpoint.
