  In this paper we describe an infinite family of Cameron-Liebler line classes
of ${\rm PG}(3,q)$ with parameter $(q^2 + 1)/2$, $q\equiv 1\pmod{4}$. The
example obtained admits ${\rm PGL}(2,q)$ as an automorphism group and it is
shown to be isomorphic to none of the infinite families known so far whenever
$q \ge 9$.
