A switchable, hydraulic damping mount includes a work chamber filled with hydraulic fluid and a compensation chamber connected to the work chamber via a channel. A partition wall mutually separates the chambers. A ferromagnetic diaphragm is arranged in the partition wall and is deflectable. An electromagnetic switching actuator has a deenergized state and an energized state and controls the diaphragm. The switching actuator applies a magnetic holding force to the diaphragm and fixes the diaphragm in a rest position when the switching actuator is in the deenergized state and reduces the magnetic holding force to such an extent that the diaphragm is enabled for movement in the longitudinal direction of the mount when the switching actuator is in the energized state.