In this paper we introduce a new algorithm for the k-Shortest Simple Paths (k-SSP) problem with an asymptotic running time matching the state of the art from the literature. It is based on a black-box algorithm due to Roddity and Zwick that solves at most 2k instances of the Second Shortest Simple Path (2-SSP) problem without specifying how this is done. We fill this gap using a novel approach: we turn the scalar 2-SSP into instances of the Biobjective Shortest Path problem. Our experiments on grid graphs and on road networks show that the new algorithm is very efficient in practice.