Consider the complete bipartite graph on $n+n$ vertices where the edges are equipped with i.i.d. exponential costs. A matching of the vertices is stable if it does not contain any pair of vertices where the connecting edge is cheaper than both matching costs. There exists a unique stable matching obtained by iteratively pairing vertices with small edge costs. We show that the total cost $C_{n,n}$ of this matching is of order $\log n$ with bounded variance, and that $C_{n,n}-\log n$ converges to a Gumbel distribution. We also show that the typical cost of an edge in the matching is of order $1/n$, with an explicit density on this scale, and analyze the rank of a typical edge. These results parallel those of Aldous for the minimal cost matching in the same setting. We then consider the sensitivity of the matching and the matching cost to perturbations of the underlying edge costs. The matching itself is shown to be robust in the sense that two matchings based on largely identical edge costs will have a substantial overlap. The matching cost however is shown to be noise sensitive, as a result of the fact that the most expensive edges will with high probability be replaced after resampling. Our proofs also apply to the complete (unipartite) graph and the results in this case are qualitatively similar.