An uncountable union of line segments with null two-dimensional measure

In this paper we construct an uncountable union of line segments $T$ which has full intersection with the sets $(\{0\}\times [0,1])\cup (\{1\}\times [0,1])\subset {\mathbb{R}}^2$ but has null two-dimensional measure. Further results are proved on the decay rate of $\mu (T)$ if the line segments comprising $T$ are replaced with increasingly fine approximations by parallelograms.