A tool body including a damping apparatus for damping vibrations in a machining tool, which machining tool comprises the tool body and a member with at least one cutting edge, wherein the damping apparatus is arranged in an internal room of the tool body and comprises • a damper mass body with an axial through bore , • a central tube , which is rigidly connected to a tool body fixed part and extends through the through bore while leaving a cavity that surrounds the tube between the tube and the through bore such that the central tube and the damper mass body are separated, wherein the cavity contains a damping fluid, • an elastic member at each axial end of the damper mass body configured to allow the damper mass body to dampingly interact with a tool body fixed part and having a central hole through which the central tube extends, and • a couple of two rigid disc-like members (17-20) for each elastic element for clamping this element therebetween and which members connect, by one axially directed side facing away from the elastic element, to the damper mass body and by the other to a tool body fixed part, wherein • each disc-like member has a central hole aligned with the central hole of the elastic element, characterized in that • each elastic element is provided with two socket-like portions , each extending axially through the central hole of a said disc-like member, wherein • the socket-like portions project in a rest state of the elastic element axially beyond the respective said axially directed side of the respective disc-like member and which portions are in an assembled state compressed by bearing under pretention onto the damper mass body and the tool body fixed part , respectively, and by that seal said cavity with respect to the exterior. • a damper mass body with an axial through bore , • a central tube , which is rigidly connected to a tool body fixed part and extends through the through bore while leaving a cavity that surrounds the tube between the tube and the through bore such that the central tube and the damper mass body are separated, wherein the cavity contains a damping fluid, • an elastic member at each axial end of the damper mass body configured to allow the damper mass body to dampingly interact with a tool body fixed part and having a central hole through which the central tube extends, and • a couple of two rigid disc-like members (17-20) for each elastic element for clamping this element therebetween and which members connect, by one axially directed side facing away from the elastic element, to the damper mass body and by the other to a tool body fixed part, • each disc-like member has a central hole aligned with the central hole of the elastic element, • each elastic element is provided with two socket-like portions , each extending axially through the central hole of a said disc-like member, wherein • the socket-like portions project in a rest state of the elastic element axially beyond the respective said axially directed side of the respective disc-like member and which portions are in an assembled state compressed by bearing under pretention onto the damper mass body and the tool body fixed part , respectively, and by that seal said cavity with respect to the exterior. <EOS>