Given a graph and a pair of terminals $s$, $t$, the next-to-shortest path problem asks for an $s\!\to \!t$ (simple) path that is shortest among all not shortest $s\!\to \!t$ paths (if one exists). This problem was introduced in 1996, and soon after was shown to be NP-complete for directed graphs with non-negative edge weights, leaving open the case of positive edge weights. Subsequent work investigated this open question, and developed polynomial-time algorithms for the cases of undirected graphs and planar directed graphs. In this work, we resolve this nearly 30-year-old open problem by providing an algorithm for the next-to-shortest path problem on directed graphs with positive edge weights.