We design exact algorithms for the following two problems in survivable network design: (i) designing a minimum cost network with a desired value of edge connectivity, which is called Minimum Weightλ-connected Spanning Subgraph and (ii) augmenting a given network to a desired value of edge connectivity at a minimum cost which is called Minimum Weightλ-connectivity Augmentation. It is easy to see that a minimum solution to these problems contains at most 2 λ(n- 1) edges. Using this fact one can design a brute-force algorithm which runs in time 2 O(λnlogn), however no better algorithms were known previously. In this paper, we give the first single exponential time algorithm for these problems, i.e. running in time 2 O(λn), for both undirected and directed networks. Our results are obtained via well known characterizations of λ-connected graphs, their connections to linear matroids and the recently developed technique of dynamic programming with representative sets. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.