Trusted AI literature to date has focused on the trust needs of users who knowingly interact with discrete AIs. Conspicuously absent from the literature is a rigorous treatment of public trust in AI. We argue that public distrust of AI originates from the under-development of a regulatory ecosystem that would guarantee the trustworthiness of the AIs that pervade society. Drawing from structuration theory and literature on institutional trust, we offer a model of public trust in AI that differs starkly from models driving Trusted AI efforts. We describe the pivotal role of externally auditable AI documentation within this model and the work to be done to ensure it is effective, and outline a number of actions that would promote public trust in AI. We discuss how existing efforts to develop AI documentation within organizations---both to inform potential adopters of AI components and support the deliberations of risk and ethics review boards---is necessary but insufficient assurance of the trustworthiness of AI. We argue that being accountable to the public in ways that earn their trust, through elaborating rules for AI and developing resources for enforcing these rules, is what will ultimately make AI trustworthy enough to be woven into the fabric of our society.