As has been known since the 90s, there is an integrable structure underlying two-dimensional gravity theories. Recently, two-dimensional gravity theories have regained an enormous amount of attention, but now in relation with quantum chaos—superficially nothing like integrability. In this paper, we return to the roots and exploit the integrable structure underlying dilaton gravity theories to study a late time, large e SBH double scaled limit of the spectral form factor. In this limit, a novel cancellation due to the integrable structure ensures that at each genus g the spectral form factor grows like T 2g+ 1, and that the sum over genera converges, realising a perturbative approach to the late-time plateau. Along the way, we clarify various aspects of this integrable structure. In particular, we explain the central role played by ribbon graphs, we discuss intersection theory, and we explain what the relations with dilaton gravity and matrix models are from a more modern holographic perspective.