A weighted undirected network is ∆ growth-bounded if the number of nodes at distance 2r around any given node is at most ∆ times the number of nodes at distance r around the node. Given a weighted undirected network with arbitrary node names and > 0, we present a routing scheme that routes along paths of stretch 1 + and uses with high probability only O(1 O(log ∆) log5 n) bit routing tables per node. Categories and Subject Descriptors C.2.1 [Computer-Communication Networks]: Network Architecture and Design—Distributed networks; G.2.2 [Discrete Mathematics]: Graph Theory—Network problems, Graph labeling. General Terms Algorithms, Theory. Keywords Compact Routing.