A linear ball bearing for longitudinally movable mounting of a shaft-like guide rail in a part surrounding the guide rail, which has a sleeve-like cage with recesses arranged one behind the other in the circumferential direction holding ball circuits which have at least one supporting section with supporting balls which are supported such that they can roll on the guide rail. The supporting balls are also supported such that they can roll on a raceway of a supporting race element inserted indirectly or directly into the cage. The supporting race element has at least one radial outer surface, arched outward at least in one direction, for contact against an inner surface of the part which surrounds the guide rail. The outer surface is formed by at least one web-like projection extending in the longitudinal direction of the supporting race element.