  A biperiodic planar network is a pair $(G,c)$ where $G$ is a graph embedded
on the torus and $c$ is a function from the edges of $G$ to non-zero complex
numbers. Associated to the discrete Laplacian on a biperiodic planar network is
its spectrum: a triple $(C,S,\nu)$, where $C$ is a curve and $S$ is a divisor
on it. We give a complete classification of networks (modulo a natural
equivalence) in terms of their spectral data. The space of networks has a large
group of cluster automorphisms arising from the $Y-\Delta$ transformations. We
show that the spectrum provides action-angle coordinates for the discrete
cluster integrable systems defined by these automorphisms.
