Democratization of AI means not only that people can freely use AI, but also that people can collectively decide how AI is to be used. In particular, collective decision-making power is required to redress the negative externalities from the development of increasingly advanced AI systems, including degradation of the digital commons and unemployment from automation. The rapid pace of AI development and deployment currently leaves little room for this power. Monopolized in the hands of private corporations, the development of the most capable foundation models has proceeded largely without public input. There is currently no implemented mechanism for ensuring that the economic value generated by such models is redistributed to account for their negative externalities. The citizens that have generated the data necessary to train models do not have input on how their data are to be used. In this work, we propose that a public data trust assert control over training data for foundation models. In particular, this trust should scrape the internet as a digital commons, to license to commercial model developers for a percentage cut of revenues from deployment. First, we argue in detail for the existence of such a trust. We also discuss feasibility and potential risks. Second, we detail a number of ways for a data trust to incentivize model developers to use training data only from the trust. We propose a mix of verification mechanisms, potential regulatory action, and positive incentives. We conclude by highlighting other potential benefits of our proposed data trust and connecting our work to ongoing efforts in data and compute governance.