This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a full-state feedback control law, an observer design, and the associated output feedback control law of both Neumann and Dirichlet boundary actuations via the backstepping method. Also, the state-feedback control law is provided when a Robin boundary input is considered. The designed observer allows the estimation of the temperature profile based on the available measurements of liquid-phase length and the heat flux at the interface. The associated output feedback controller ensures the global exponential stability of the estimation errors, the H-1-norm of the distributed temperature, and the moving interface at the desired setpoint under some explicitly given restrictions on the setpoint and observer gain.