  This paper investigates a certain 2-Calabi-Yau triangulated category D whose
Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting
subcategories of D form a so-called cluster structure, and we classify these
subcategories in terms of what one may call `triangulations of the
infinity-gon'. This is reminiscent of the cluster category C of type A_n which
is a 2-Calabi-Yau triangulated category whose Auslander-Reiten quiver is a
quotient of ZA_n. The cluster tilting subcategories of C form a cluster
structure and they are classified in terms of triangulations of the (n+3)-gon.
The category D behaves like a `cluster category of type A_{\infty}'.
