Let
Γ
\Gamma
be a countable abelian group. An (abstract)
Γ
\Gamma
-system
X
\mathrm {X}
- that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of
Γ
\Gamma
- is said to be a Conze–Lesigne system if it is equal to its second Host–Kra–Ziegler factor
Z
2
(
X
)
\mathrm {Z}^2(\mathrm {X})
. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian
Γ
\Gamma
, namely that they are the inverse limit of translational systems
G
n
/
Λ
n
G_n/\Lambda _n
arising from locally compact nilpotent groups
G
n
G_n
of nilpotency class
2
2
, quotiented by a lattice
Λ
n
\Lambda _n
. Results of this type were previously known when
Γ
\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
)
U^3(G)
norm for arbitrary finite abelian groups
G
G
.