We define the problem of unicast routing subject to delay constraints in point-to-point connection-oriented networks as a delay-constrained least-cost path problem. This problem is NP-complete, and therefore we propose a simple, efficient, distributed heuristic solution: the delay-constrained unicast routing (DCUR) algorithm. DCUR requires limited information about the network state to be kept at each node. This information is stored in a cost vector and a delay vector which resemble the distance vectors of some existing routing protocols. We prove the correctness of DCUR by showing that it is always capable of constructing a loop-free delay-constrained path within finite time, if such a path exists. The number of computations DCUR performs at each node participating in the path construction process is fixed, irrespective of the size of the network. The message complexity of DCUR is O(jV j 3) messages in the worst case, where jV j is the number of nodes in the network. However, simulation results show that, on the average, much fewer messages are required. Therefore, DCUR scales well to large networks. We also use simulation to compare DCUR to the optimal delay-constrained least-cost path algorithm, and to the least-delay path algorithm. Our results show that DCUR yields satisfactory performance with respect to both path cost and path delay.