La2Mo2O9 (LAMOX) is a fast oxygen ion conductor which shows high oxygen ion conductivities comparable to those of yttria-sabilized zirconia (YSZ). LAMOX is subject to a structural phase transition from the nonconductive monoclinic form to the highly conductive cubic form at about 580 degrees C. The origin of the conductivity in cubic LAMOX has been suggested to be due to a "disorder" in the O sublattice without any insight into the real distribution of the oxygen ion. In this paper, thanks to the application of the neutron atomic pair distribution function (PDF) analysis, we provide evidence that the local structure of the cubic polymorph of LAMOX is exactly the same of that of the monoclinic phase, thus indicating that the structural phase transition is actually a transition from a static to a dynamic distribution of the oxygen defects. This work represents the first application of the atomic-pair distribution function analysis to the study of an oxygen fast-oxide ion conductor and clearly indicates that a more reliable and detailed description of their local structure, particularly in the highly conductive phases, can lead to a better comprehension of the structure-property correlation, which is the starting point for the design of new and optimized functional materials.