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Zassenhaus conjectured that any unit of finite order in the integral group ring $\\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\\mathbb{Q}G$ to an element of the form $\\pm g$ with $g \\in G$. Though known for some series of solvable groups, the conjecture has been proved only for thirteen non-abelian simple groups. We prove the Zassenhaus Conjecture for the groups $\\operatorname{PSL}(2,p)$, where $p$ is a Fermat or Mersenne prime. 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