  We prove that the product of an Enriques surface and a very general curve of
genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles.
This provides the first examples of smooth projective complex threefolds of
Kodaira dimension zero for which the integral Hodge conjecture fails, and the
first examples of non-algebraic torsion cohomology classes of degree 4 on
smooth projective complex threefolds.
