The purpose of this paper is to establish a geometric scattering result for a conformally invariant nonlinear wave equation on an asymptotically simple space-time. The scattering operator is obtained via trace operators at null infinities. The proof is achieved in three steps. A priori linear estimates are obtained via an adaptation of the Morawetz vector field in the Schwarzschild space-time and a method used by Hörmander for the Goursat problem consisting in writing conformal null infinity as a graph. A well-posedness result for the characteristic Cauchy problem on a light cone at infinity for small data is then obtained. This requires uniform Sobolev estimates on a timelike foliation coming from a control of the norms of extension operators and Sobolev embeddings. Finally, the trace operators on conformal infinities are built and used to define the conformal scattering operator.