  H.J. 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. This increases the list of non-abelian simple groups for which the
conjecture is known by probably infinitely many, but at least by 49, groups.
Our result is an easy consequence of known results and our main theorem which
states that the Zassenhaus Conjecture holds for a unit in
$\mathbb{Z}\operatorname{PSL}(2,q)$ of order coprime with $2q$, for some prime
power $q$.
