A diaphragm for a pump has a one-piece elastomeric body centered on an axis and formed with an annular outer clamping edge, an annular flexible web extending radially inward from the outer edge, and a core joined by the web to the outer edge and formed in turn by an upper wall and a lower wall axially spaced therefrom. This lower wall is formed on the axis with a throughgoing hole. An insert between the walls extends through the hole, and interengaging formations on the insert and on the lower wall radially couple the insert to the lower wall at an inner edge of the hole.