A consecutive-2-out-of-n cycle line is a system of n items ordered into a cycle line such that the system fails if and only if two consecutive items both fail. A double-loop ring network for computers is such a cyclic system when the computers are also connected by a second loop which skips every other computer in the cycle if n is even, there are two half loops. Suppose that item Ii works with probability Pi and that the items have been indexed so that P1 ≤ P2 ≤... ≤ Pn. Suppose further that any permutation of the n items constitutes a system. It has been conjectured that the cyclic system $$C_n^{\ast} = I_nI_1I_{n-1}I_3I_{n-3} \ldots I_{n-4}I_4I_{n-2}I_2I_n$$ minimizes the probability of failure over all such arrangements and that the line system Ln* = I1InI3In-2... In-3I4In-1I2 minimizes the probability of failure over all arrangements of the items into a line the line conjecture follows from the cycle conjecture by setting Pn = 1. In this paper we prove the cycle conjecture.