A rotary cage press provided internally with a deformable flexible membrane having the form of a half of the cage, this press being characterized in that the elements of recovering theliquid part of the matter include on the cage, at least one external racking device, and a connection unit through which opens out the outlet orifice of the racking device; and on the chassis, a liquid taking device including a telescopic tubular unit allowing a retracted position in which the telescopic unit presents no obstacle to the rotation of the cage and an extended position in which the telescopic unit is adapted, when the cage is in a predetermined pressing angular position, to come against the connection unit bearing in a seal-tight manner around the outlet orifice of the racking device.