We investigate the nonlinear vibrations of a clamped-clamped buckled beam in the case of a one-to-one internal resonance between the first and second modes when one of them is externally excited by a primary resonance. To examine whether these two modes are nonlinearly coupled, we use the method of multiple scales to directly attack the partial-differential equation and associated boundary conditions and obtain the equations governing the modulation of their amplitudes and phases. We find that the two modes are nonlinearly coupled. To investigate the large-amplitude dynamics, we use a multi-mode Galerkin discretization to reduce the partial-differential equation, in space and time, governing the nonlinear vibrations of the buckled beam into a set of nonlinearly coupled ordinary-differential equations in time only. We use a shooting method to compute periodic orbits of the discretized equations and Floquet theory to investigate the stability of these solution and their bifurcations. We report theoretically and experimentally an energy transfer from the first mode, which is externally excited by a primary resonance, to the second mode.