The European Parliament has adopted a proposal to explore the impact of a legal personhood category for AI systems, comparable to corporate personhood ("AI Personhood"). We propose that it is premature to introduce AI Personhood, primarily because (i) the scope of AI is still ill-defined, (ii) the potential economic efficiencies and distribution of gains is uncertain, (iii) the ability of existing legal structures to achieve similar ends have not been sufficiently analyzed, (iv) the moral requirements for personhood have not yet been met, and (v) it is not yet possible to assess the social concerns arising from AIs that are indistinguishable from humans. To support our conclusion, we discuss (1) the relevance of legal personhood, (2) the definitional difficulties surrounding AI, (3) currently applicable legal principles, (4) the potential benefits and drawbacks of AI Personhood, and (5) the conditions that might justify such a category in the future. We propose four specific necessary conditions---technological, economic, legal, and moral---for AI Personhood, but observe that the conditions have not yet been met and seem unlikely to be met soon.