  We prove that there exists a pencil of Enriques surfaces defined over
$\mathbb{Q}$ with non-algebraic integral Hodge classes of non-torsion type.
This gives the first example of a threefold with the trivial Chow group of
zero-cycles on which the integral Hodge conjecture fails. As an application, we
construct a fourfold which gives the negative answer to a classical question of
Murre on the universality of the Abel-Jacobi maps in codimension three.
