Summary: We consider the resolvent \((\lambda-a)^{-1}\) of any \(\mathcal R\)-diagonal operator \(a\) in a \(\Pi_1\)-factor. Our main theorem (Theorem 1.1) gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the \(\mathcal R\)-transform of the operator \(|\lambda-c|^2\), where \(c\) is Voiculescu's circular operator, and we give an asymptotic formula for the negative moments of \(|\lambda-c|^2\) for any \(\mathcal R\)-diagonal \(a\). We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce partition structure diagrams in Section 4, a new combinatorial structure arising in free probability.