Cameron-Liebler line classes were introduced in \cite{CL}, and motivated by a question about orbits of collineation groups of $\PG(3,q)$. These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets. In this paper we construct an infinite family of Cameron-Liebler line classes in $\PG(3,q)$ with new parameter $x=(q+1)^2/3$ for all prime powers $q$ congruent to 2 modulo 3. The examples obtained when $q$ is an odd power of two represent the first infinite family of Cameron-Liebler line classes in $\PG(3,q)$, $q$ even.