Based on the regular and predictable features of topology changes in low earth orbit (LEO) satellite networks, a dynamic source routing algorithm, namely adaptive route-selection (ARS) algorithm, was presented for LEO satellite networks. In the algorithm, logical location is adopted to hide satellite mobility from the routing algorithm so that a source satellite can compute the path with minimum propagation delay between source and destination satellites by only knowing their logical locations, which avoids the exchange overhead of collecting route information from other intermediate satellites. An efficient metric was proposed to denote the path with minimum propagation delay on the basisof its distribution properties. Then the path is designated by source satellite in packet header and according to the path the packet is forwarded to its destination satellite by intermediate satellites, which can greatly decrease the route overhead compared with other dynamic source routing algorithms. The algorithm can also deal with packet routing issues in case of link congestion or satellite failure. The performance comparison of the ARS algorithm with datagram routing algorithm (DRA) and Bellman′s shortest path (SP) algorithm was evaluated by simulation. The simulation results show that the proposed algorithm can guarantee the demand of end-to-end transmission delay for data packets with lower computation and exchange overhead.