Let G be an n-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of G with minimum weight and (2) a cycle of G with minimum weight. The best previously known algorithm for the former problem, running in O(n log 3 n) time, can be obtained from the algorithm of Łącki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the O(n log 3 n)-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in O(n log n log log n) time via a divide-and-conquer algorithm that finds a shortest non-degenerate cycle. The kernel of our result is an O(n log log n)-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph.