Device for compensating the volume of a damping liquid for a damper, used in particular for suspension assemblies for cycles and other vehicles, comprising - at least one hollow cylindrical main body containing a fluid and having at least one end provided with an axial passage for the passage, sealing and guiding of a rod, said rod being rigidly connected to a main piston which moves in translation inside said main body and divides it into two different work chambers, one constituting a compression chamber and the other constituting an expansion chamber, the compression chamber and the expansion chamber being directly connected to each other via one or more internal channels in the piston for the passage of the fluid through said piston and one or more compensation ports at the piston opening into the compression chamber and into the expansion chamber, - valves for closing off the compensation ports opening into the compression chamber and into the expansion chamber, - at least one rigid part interposed between the closing-off valves opening into the compression chamber and into the expansion chamber, said rigid part being able to move in translation in the axial direction of the rod and the main body between a first end position in which the compensation ports opening into the expansion chamber are closed off by said dedicated valve(s), while the compensation ports opening into the compression chamber are not closed off by said dedicated valve(s), and a second end position, referred to as the compression position, in which the compensation ports opening into the compression chamber are closed off by said dedicated valve(s), while the compensation ports opening into the expansion chamber are not closed off by said dedicated valve(s), the compensation ports and the rigid part being positioned such that it is impossible to simultaneously close both the compensation ports opening into the compression chamber and the compensation ports opening into the expansion chamber, and such that the sum of the cross-sections of the compensation ports which are not closed off by the rigid actuating part still allows for a direct passage of oil, characterized in that said rigid part is arranged to move in translation freely through a passage provided at the piston, which passage is different from said internal channel(s). - at least one hollow cylindrical main body containing a fluid and having at least one end provided with an axial passage for the passage, sealing and guiding of a rod, said rod being rigidly connected to a main piston which moves in translation inside said main body and divides it into two different work chambers, one constituting a compression chamber and the other constituting an expansion chamber, the compression chamber and the expansion chamber being directly connected to each other via one or more internal channels in the piston for the passage of the fluid through said piston and one or more compensation ports at the piston opening into the compression chamber and into the expansion chamber, - valves for closing off the compensation ports opening into the compression chamber and into the expansion chamber, - at least one rigid part interposed between the closing-off valves opening into the compression chamber and into the expansion chamber, said rigid part being able to move in translation in the axial direction of the rod and the main body between a first end position in which the compensation ports opening into the expansion chamber are closed off by said dedicated valve(s), while the compensation ports opening into the compression chamber are not closed off by said dedicated valve(s), and a second end position, referred to as the compression position, in which the compensation ports opening into the compression chamber are closed off by said dedicated valve(s), while the compensation ports opening into the expansion chamber are not closed off by said dedicated valve(s), the compensation ports and the rigid part being positioned such that it is impossible to simultaneously close both the compensation ports opening into the compression chamber and the compensation ports opening into the expansion chamber, and such that the sum of the cross-sections of the compensation ports which are not closed off by the rigid actuating part still allows for a direct passage of oil, characterized in that said rigid part is arranged to move in translation freely through a passage provided at the piston, which passage is different from said internal channel(s).