New constraints for non-Newtonian gravity in the nanometer range from the improved precision measurement of the Casimir force

We obtain constraints on non-Newtonian gravity following from the improved precision measurement of the Casimir force by means of an atomic force microscope. The hypothetical force is calculated in an experimental configuration (a sphere above a disk both covered by two metallic layers). The strengthenings of constraints up to 4 times comparing the previous experiment and up to 560 times comparing the Casimir force measurements between dielectrics are obtained in the interaction range 5.9 nm<~λ<~115 nm. Recent speculations about the presence of some unexplained attractive force in the considered experiment are shown to be unjustified.