Abstract: The apparatus allows providing a rim on the tips for ballpoint pens even if made of a very hard material, such as stainless steel, as it can withstand very great pressure. It comprises a first body (1) rotating about an axis (8) which coincides with the axis of the tip (10) to be provided with a rim. The body (1) is provided with an eccentric hole (11) on whose axis (9) a second body (3) rotates on first rolling bearings, in which second body (3) a pivot (5) rotates coaxially on second rolling bearings. The pivot (5) contacts on its inner end (5") a pushing member (12) and on its external end a first convergent conical surface (5') which presses directly on the edge (6) of the tip to be provided with a rim with the aid of a second divergent conical surface (3' ) joined to the second body (3). The two surfaces (3'and 5') are such as to constitute the angle .alpha. which is most suitable for providing a rim on the edge (6) of the tip (10) by retaining the ball with the desired clearance.