We discover a strong localization of flexural (bi-Laplacian) waves in rigid thin plates. We show that clamping just one point inside such a plate not only perturbs its spectral properties, but essentially divides the plate into two independently vibrating regions. This effect progressively appears when increasing the plate eccentricity. Such a localization is qualitatively and quantitatively different from the results known for the Laplacian waves in domains of irregular boundary. It would allow us to control the confinement of mechanical vibrations in rigid plates and of eddies in the slow Stokes flow.