We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where 3g−3+n≥2. We show that every C^1-diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden's Theorem that the isometry group of the Teichmüller metric is the extended mapping class group. 

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