The authors develop a distributed algorithm for finding a shortest-path tree rootes at any node of a plane network with positive edge costs and prove its correctness. The algorithm is of message and time complexity \(O(pn)\) in the worst case, where \(p\) is the smallest number of faces needed to cover all nodes, taken over all possible plane embeddings of the network, and \(n\) is the number of nodes. The proposed algorithm contains two principal parts: an outerplanar decomposition of a \(p\)-plane graph and a distributed single-source shortest-path algorithm for biconnected outerplanar graphs. The existence of an outerplanar decomposition is proved and a distributed algorithm for identifying such a decomposition is presented. Since the single-source shortest-path algorithm for outerplanar graphs proposed in this paper could be fooled in the case when the edge cost exceeds the distance between its endpoints, the authors show how to handle this problem. An interesting adaptive routing scheme for dynamic \(p\)-plane graphs is presented as an application of the theory developed in the paper.