Artificial intelligence (AI) is rapidly becoming a global infrastructure for cognition, coordination, and coercion. Yet the policy tools we are using to govern its cross-border impacts: \emph{science diplomacy, tech diplomacy, and classical arms control,} were designed for a different technological era. They treat AI either as a sectoral topic, a bundle of discrete risks, or a generic ``emerging technology'' to be slotted into existing regimes. This paper argues that such inherited tools are necessary but insufficient. It develops a framework for \emph{AI diplomacy} that is explicitly tailored to AI's infrastructural character. The core contribution is a four-by-four framework organised around four \emph{layers} of AI governance targets (resources, systems, applications, epistemics) and four \emph{diplomatic functions} (coordination, containment, composition, contestation). The framework surfaces three hard problems that define the frontier of AI diplomacy: (a) epistemic governance and the challenge of building something like an ``IPCC for AI'' in a fast-moving, private-sector-led domain; (b) the status of AI systems as \emph{quasi-actors} and the problem of legitimate delegation of diplomatic and coercive discretion; and (c) justice and composition for the Global South in a world where compute, data, and talent are highly concentrated. These are not simply descriptive gaps; they constitute a research and policy agenda. The final section briefly applies the framework to Canada as an illustrative case. As a middle power with strong AI research hubs, a bilingual and multicultural society, and a tradition of functional multilateralism, Canada is well placed to pursue an AI diplomacy agenda that leverages coordination and composition rather than relying on hard power. The Canadian case illustrates how the framework can guide the strategic positioning of actors that are neither AI superpowers nor rule-takers.