We study a model for a nonsingular cosmic bounce in N=1 supergravity, based on supergravity versions of the ghost condensate and cubic Galileon scalar field theories. The bounce is preceded by an ekpyrotic contracting phase, which prevents the growth of anisotropies in the approach to the bounce and allows for the generation of scale-invariant density perturbations that carry over into the expanding phase of the Universe. We present the conditions required for the bounce to be free of ghost excitations, as well as the tunings that are necessary in order for the model to be in agreement with cosmological observations. All of these conditions can be met. Our model thus provides a proof-of-principle that nonsingular bounces are viable in supergravity, despite the fact that during the bounce the null energy condition is violated.