  Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system
$\mathrm{X}$ - that is, an (abstract) probability space equipped with an
(abstract) probability-preserving action of $\Gamma$ - is said to be a
Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor
$\mathrm{Z}^2(\mathrm{X})$. The main result of this paper is a structural
description of such Conze-Lesigne systems for arbitrary countable abelian
$\Gamma$, namely that they are the inverse limit of translational systems
$G_n/\Lambda_n$ arising from locally compact nilpotent groups $G_n$ of
nilpotency class $2$, quotiented by a lattice $\Lambda_n$. Results of this type
were previously known when $\Gamma$ was finitely generated, or the product of
cyclic groups of prime order. In a companion paper, two of us will apply this
structure theorem to obtain an inverse theorem for the Gowers $U^3(G)$ norm for
arbitrary finite abelian groups $G$.
