Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical decompositions of arbitrary graphs, one based on strength and the other on arboricity. Here we introduce an exact-arithmetic algorithm for spanning tree modulus and the strength-based decomposition using Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between spanning tree modulus and critical edge sets from the vulnerability problem. This paper introduces the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests.