This Letter presents the first direct experimental measurements of the damping of toroidicity-induced Alfven eigenmodes (TAE), carried out in the JET tokamak. These measurements were obtained during the first experiments to drive these modes with antennas external to a tokamak plasma. Different regimes corresponding to different dominant TAE absorption mechanisms with a wide range of damping rates, 10(-3) less than or equal to gamma/omega less than or equal to 10(-1), have been identified in Ohmically heated plasma discharges.