An apparatus for producing a low-PMD optical fiber having a furnace for melting a lower portion of an optical preform; a traction device for pulling an optical fiber from the lower portion of an optical preform; a spinning device for imparting a substantially constant and unidirectional spin to the optical fiber as it is pulled, which causes the fiber to undergo an elastic torsion; a winding device for winding the optical fiber onto a reel; and a twisting device for imparting to the spun optical fiber a unidirectional twist in a direction opposite the elastic torsion, so as to control the residual twist in the optical fiber. A process for producing the fiber, an optical fiber and a cable are also provided.