Femtosecond time-resolved core-level photoemission spectroscopy with a free-electron laser is used to measure the atomic-site specific charge-order dynamics of the charge-density wave in the Mott insulator 1T-TaS2. After strong photoexcitation, a prompt loss of charge order and subsequent fast equilibration dynamics of the electron-lattice system are observed. On the time scale of electron-phonon thermalization, about 1 ps, the system is driven across a phase transition from a long-range charge ordered state to a quasiequilibrium state with domainlike short-range charge and lattice order. The experiment opens the way to study the nonequilibrium dynamics of condensed matter systems with full elemental, chemical, and atomic-site selectivity.