In this paper, we study the problem of multi-path routing with bandwidth and delay constraints, which arises in applications for video delivery over bandwidth-limited networks. Assume that each link in the network has a bandwidth and a delay. For a given source-destination pair and a bandwidth requirement, we want to find a set of source to destination paths such that the delay of the longest path is minimized while the aggregated bandwidth of the set of paths meets the bandwidth requirement. This problem is NP-hard, and the state of the art is a maximum flow based heuristic. We first construct a class of examples showing that the maximum flow based heuristic could have very bad performance. We then present a fully polynomial time approximation scheme that can compute a (1 +)-approximation with running time bounded by a polynomial in 1/and the input size of the instance. Given the NP-hardness of the problem, our approximation scheme is the best possible. We also present numerical results confirming the advantage of our scheme over the current state of the art.