Self-stabilization for non-masking fault-tolerant distributed system has received considerable research interest over the last decade. In this paper, we propose a self-stabilizing algorithm for 2-edge-connectivity and 2-vertex-connectivity of an asynchronous distributed computer network. It is based on a self-stabilizing depth-first search, and is not a composite algorithm in the sense that it is not composed of a number of self-stabilizing algorithms that run concurrently. The time and space complexities of the algorithm are the same as those of the underlying self-stabilizing depth-first search algorithm which are O(dnΔ) rounds and O(n\log Δ) bits per processor, respectively, where Δ(<= n) is an upper bound on the degree of a node, d (<= n) is the diameter of the graph, and n is the number of nodes in the network.