We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in undirected planar graphs both min-cut and min $st$-cut have $O(n \log \log n)$ solutions, in directed planar graphs our result makes min-cut faster than min $st$-cut, which currently requires $O(n \log n)$.