This paper studies the H-infinity control problem for a nonlinear, unbounded, infinite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi-Isaacs (HJI) equation, proving that the H-infinity problem can be solved if and only if the HJI equation has a positive definite viscosity supersolution, vanishing and continuous at the origin. In order to do so, the standard definition of the H-infinity problem has to be relaxed by using the theory of differential games. We apply our results to the one phase Stefan problem.