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Am. Math. Soc. 375 (2022), no. 1, 627\u2013668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let  and let  be divisible by . Then, in the random \u2010uniform hypergraph process on  vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph , the random graph process contains a \u2010factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. 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