  Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the
compositum of all cubic extensions of $\mathbb{Q}$. In this article we show
that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine
20 possibilities for its structure, along with a complete description of the
$\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into
each case. We provide rational parameterizations for each of the 16 torsion
structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism
classes of elliptic curves, and a complete list of $j$-invariants for each of
the 4 that do not.
