We report the implementation of a highly sensitive tunable beam displacer based on the concept of weak value amplification, that allows to displace the centroid of a Gaussian beam a distance much smaller than its beam width without the need to deflect the direction of propagation of the input beam with movable optical elements. The beam’s centroid position can be displaced by controlling the linear polarization of the output beam, and the dependence between the centroid’s position and the angle of polarization is linear.