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The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been conjectured that both the spin and the loop O(n) models exhibit exponential decay of correlations when n>2. We verify this for the loop O(n) model with large parameter n, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight x. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when nx6 is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when nx6 is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.