In this paper, we propose a truthful combinatorial auction for the joint radio and processing resource allocation problem in the context of a Cloud-based Radio Access Network (C-RAN). We formulate the auction as an Integer Linear Program (ILP), taking into accurate account interference constraints while leveraging radio resource reuse to generate an optimal revenue for the RAN operator. Then, we propose Truthful Greedy Approach (TGA), an effective and truthful heuristic that guarantees a close-to-optimum revenue compared to the one obtained with the ILP formulation. Extensive simulations, conducted in representative network scenarios, compare and evaluate our auction with state-of-the-art approaches from the literature, showing its effectiveness.