Summary: We calculate the graded center of the stable category of a Brauer tree algebra. The canonical map from the Tate analogue of Hochschild cohomology to the graded center of the stable category is shown to induce an isomorphism modulo taking quotients by suitable nilpotent ideals. More precisely, we show that this map is surjective with nilpotent kernel in even degrees, while this map need not be surjective in odd degrees in general.