Under the technical condition of separability of some topological space of cohomology, we show that the Godbillon-Vey number of a foliation of codimension one on a compact orientable 3-fold is topologically rigid if and only if the foliation admits a projective transversal structure. Here by the rigidity of the Godbillon-Vey number we mean that it is constant under the infinitesimal singular deformations of the foliation, and a foliation admits a transversal projective structure whenever it can be glued from the level sets of locally defined functions related by fractional linear transition maps.