A nonsingular transformation is said to be doubly ergodic
if for all sets $A$ and $B$ of positive measure there exists
an integer $n>0$ such that $\lambda(T^{-n}(A)\cap A)>0$ and
$\lambda(T^{-n}(A)\cap B)>0$. While double ergodicity is
equivalent to weak mixing for finite measure-preserving
transformations, we show that this is not the case for
infinite measure preserving transformations. We show that all
measure-preserving tower staircase rank one constructions are
doubly ergodic, but that there exist tower staircase
transformations with non-ergodic Cartesian square. We also
show that double ergodicity implies weak mixing but that there
are weakly mixing skyscraper constructions that are not doubly
ergodic. Thus, for infinite measure-preserving
transformations, double ergodicity lies properly between weak
mixing and ergodic Cartesian square. In addition we study some
properties of double ergodicity.