Does non-zero set contain non-zero sets in each coordinate?

Question

Suppose $A\in \mathcal B([0,1]^2)$ is a Borel measurable subset of the square of the unit interval $[0,1]^2$ with positive Lebesgue measure $\text{Leb}_{[0,1]^2}(A)>0$. Does this already imply the existence of measurable sets $A_1,A_2 \in \mathcal B([0,1])$ with their cartesian product $A_1 \times A_2 \subseteq A$ contained in $A$ and having positive Lebesgue measure $\text{Leb}_{[0,1]}(A_1) >0$ and $\text{Leb}_{[0,1]}(A_2) >0$?