We propose a refinement of the maxmin approach to distributional robustness. A mechanism's payoff guarantee over an ambiguity set is robust if the guarantee is approximately satisfied at priors near the ambiguity set (in the weak topology). We show that many maxmin-optimal mechanisms in the literature give payoff guarantees that are not robust. Such mechanisms are often tailored to degenerate worst-case priors, making them simple but fragile. We propose a method of enriching the ambiguity set to include all nearby priors. The associated maxmin-optimal mechanism's payoff guarantee (over this new ambiguity set) is always robust.