Computer networks are often multi-layered. For simplicity, let us focus on two-layered networks with logical layer and physical layer. Such a network can be modeled as a labeled graph $G = (V, E)$ with a label set $L = \{\ell _{1}, \ell _{2}, {\dots }, \ell _{q} \}$ , in which each edge (denotes logical connection) $e \in E$ has a label (denotes physical link) $\ell (e)$ from L. The key issue is that different edges may have the same label. In the weighted minimum Label s-t Cut problem, we are given a labeled graph $G=(V,E)$ with label set L, where each label $\ell$ has a nonnegative weight $w_{\ell }$ , a source $s \in V$ and a sink $t \in V$ . The problem asks to find a minimum weight label subset $L'$ (called a label s-t cut) such that the removal of all edges with labels in $L'$ disconnects s and t. Label s-t cut depicts the connectivity bottleneck of a layered network. It is a natural generalization of the edge connectivity of a graph. In this paper, we provide an approximation algorithm for the weighted Label s-t Cut problem with ratio $O(n^{2/3})$ , where n is the number of vertices. This is the first approximation algorithm for the problem whose ratio is given in terms of n. The key point of the algorithm is a mechanism to interpret label weight on an edge as both its length (as in the Shortest s-t Path problem) and capacity (as in the Min s-t Cut problem). Experiments on random graphs show that the algorithm has also good practical performance.