A pressure damping device for a fluid circuit, this device comprising: - a lower body comprising fluid inlet and outlet ports (12a, 12b) which are connected to a chamber located between the inlet and outlet ports (12a, 12b), the body being formed in one single piece and the chamber comprising a first bottom, - an upper cover attached to the body above the chamber, this cover comprising an internal cavity with a second bottom oriented towards the first bottom of the chamber, - an elastically deformable membrane interposed between the body and the cover, the membrane being generally circular in shape and comprising a central portion which has a constant thickness and a peripheral portion which is clamped between the body and the cover to ensure a seal between the chamber and the cavity, - a circular cup arranged in the cavity and on the central portion of the membrane, and - a spring interposed between the cup and the second bottom of the cavity, wherein the body comprises a support member for supporting the membrane, this support member projecting from the first bottom and comprising a free upper end on which the membrane is adapted to be supported, and in which: - the member extends from the first bottom along a central axis to a height which is less than or equal to a height of the chamber, this member having a circular outline and having an external diameter D3 measured with respect to said axis, - the cup is centred on said axis and has an external diameter D10 measured with respect to said axis, with 0.8×D3 < D10 < 1.2xD3, and - the membrane is centred on said axis and its peripheral portion has a clamping internal diameter D1 measured with respect to said axis, with D1 > 1.5xD3. - a lower body comprising fluid inlet and outlet ports (12a, 12b) which are connected to a chamber located between the inlet and outlet ports (12a, 12b), the body being formed in one single piece and the chamber comprising a first bottom, - an upper cover attached to the body above the chamber, this cover comprising an internal cavity with a second bottom oriented towards the first bottom of the chamber, - an elastically deformable membrane interposed between the body and the cover, the membrane being generally circular in shape and comprising a central portion which has a constant thickness and a peripheral portion which is clamped between the body and the cover to ensure a seal between the chamber and the cavity, - a circular cup arranged in the cavity and on the central portion of the membrane, and - a spring interposed between the cup and the second bottom of the cavity, wherein the body comprises a support member for supporting the membrane, this support member projecting from the first bottom and comprising a free upper end on which the membrane is adapted to be supported, and in which: - the member extends from the first bottom along a central axis to a height which is less than or equal to a height of the chamber, this member having a circular outline and having an external diameter D3 measured with respect to said axis, - the cup is centred on said axis and has an external diameter D10 measured with respect to said axis, with 0.8×D3 < D10 < 1.2xD3, and - the membrane is centred on said axis and its peripheral portion has a clamping internal diameter D1 measured with respect to said axis, with D1 > 1.5xD3. - the member extends from the first bottom along a central axis to a height which is less than or equal to a height of the chamber, this member having a circular outline and having an external diameter D3 measured with respect to said axis, - the cup is centred on said axis and has an external diameter D10 measured with respect to said axis, with 0.8×D3 < D10 < 1.2xD3, and - the membrane is centred on said axis and its peripheral portion has a clamping internal diameter D1 measured with respect to said axis, with D1 > 1.5xD3.