A biperiodic planar electrical 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 electrical network is its spectral transform: a triple (C, S, ν), where C is a curve and S is a divisor on it, which we show is a point in the Prym variety of C. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral transform.