Given an element γ in a group γ, the stable exponent p+(γ) of γ is defined as p+(γ) =lim supn→∞P(γn) denotes the exponent of P(γn) = sup{k/ ∀γo∈ γ such that γn= γkoWe prove that if γ acts properly discontinuously by isometrics on a proper geodesic Gromov-hyperbolic metric space and γ ∈ γ is of hyperbolic type, then P+(γ) is an integer. This implies that the stable exponent of every element of infinite order in a word hyperbolic group is an integer. We also show that, in a translation discrete group, the stable exponent of every element of infinite order is finite.