The aim of the present paper is to characterize almost co-Kähler manifolds whose metrics are the Riemann solitons. At first we provide a necessary and sufficient condition for the metric of a 3-dimensional manifold to be Riemann soliton. Next it is proved that if the metric of an almost co-Kähler manifold is a Riemann soliton with the soliton vector field ξ, then the manifold is flat. It is also shown that if the metric of a (κ, µ)-almost co-Kähler manifold with κ < 0 is a Riemann soliton, then the soliton is expanding and κ, µ, λ satisfies a relation. We also prove that there does not exist gradient almost Riemann solitons on (κ, µ)-almost co-Kähler manifolds with κ < 0. Finally, the existence of a Riemann soliton on a three dimensional almost co-Kähler manifold is ensured by a proper example