An apparatus for conveying pasty substances, for example sausage stuffing, is designed as a radial displacer piston conveyor for a rotary angle-proportional delivery of substance portions. This is made possible by designing a control cam section controlling the substance discharge as a circular involute. It is proposed to reduce the large and theoretically infinitely high acceleration forces acting on each displacer piston at the instant of the start of its discharge action by designing the involute section of the control cam with a run-up section and a run-off section, each, whose courses are designed mirror-inverted to each other. The arrangement thereof is such that at the instant when one displacer piston runs onto said run-up section, the preceding piston enters the run-off section, so that, during the passage of those two cam sections, two displacer pistons, each, participate in the substance discharge. The acceleration values thereof add up to become zero.