In this paper we study the compatibility of rigidity with various notions of weak mixing in infinite ergodic theory. We prove that there exists an infinite measure-preserving transformation that is spectrally weakly mixing and rigid, but not doubly ergodic. We also construct an example to show that rigidity is compatible with rational ergodicity. At the end of the paper we explore the structure of rigidity sequences for infinite measure-preserving transformations that have ergodic Cartesian square, as well as the structure of rigidity sequences for infinite measure-preserving transformations that are rationally ergodic. All of our constructions are via the method of cutting and stacking.