This paper provides global exponential stabilization results by means of boundary feedback control for one dimensional nonlinear unstable reaction-diffusion partial differential equations (PDEs) with nonlinearities of superlinear growth. The class of systems studied are parabolic PDEs with nonlinear reaction terms that provide "damping" when the norm of the state is large (the class includes reaction-diffusion PDEs with polynomial nonlinearities). The case of Dirichlet actuation at one end of the domain is considered and a Control-Lypunov Functional construction is applied in conjunction with Stampacchia's truncation method. The paper also provides several important auxiliary results, among which is an extension of Wirtinger's inequality, used here for the construction of the Control-Lyapunov functional.