We consider a reaction-di�usion system which models a fast reversible reaction between two mobile reactants and prove convergence of the solutions as the reaction rate tends to in�nity, where the limiting problem is given by a di�usion equation with nonlinear di�usion. Since the rate function has no sign, the usual methods to obtain a priori estimates in the case of irreversible reactions do not apply; we deduce instead a priori estimates from computations based on Lyapunov function techniques.