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First prove the torsion in an HNN extension statement directly from Britton's lemma. Let $z$ have finite order in an HNN extension of $G$. Among its conjugates, choose a reduced normal-form word with the smallest possible number $k$ of stable letters. If $k=0$, that conjugate lies in the embedded base $G$, so its finite order and the base's torsion-free group property force it to be the identity.

Suppose instead that $k>0$. There are no internal pinches in an HNN extension because the word is reduced. There is also no pinch at the join of its end with its beginning. Indeed, write it as $g_0{t}^{{\epsilon }_1}{g}_1\cdots t^{{\epsilon }_k}{g}_k$ and conjugate off $g_0$ to obtain $t^{{\epsilon }_1}{g}_1\cdots t^{{\epsilon }_k}(g_k{g}_0)$. If the joined subword $t^{{\epsilon }_k}(g_k{g}_0)t^{{\epsilon }_1}$ were a pinch, its two signs would be opposite; conjugating by $t^{{\epsilon }_1}$ and using the pinch relation would reduce the stable-letter count by two, contradicting minimality. Thus every positive power is reduced with exactly $rk$ stable letters in the $r$th power. Britton's lemma makes every such power nonidentity, contradicting finite order. We have provedThe base embeds by the same lemma, so finite order has not been lost or artificially introduced by a quotient.

For the countable torsion-free embedding with two conjugacy classes, start with $G_0=G$. Given a countable torsion-free $G_j$, list all ordered pairs $(a,b)$ of its nonidentity elements. Both elements have infinite order, so the assignment $a^r\mapsto b^r$ is an isomorphism between their infinite cyclic subgroups. Successively adjoin one stable letter for each pair with relation $t_{a,b}^{-1}a{t}_{a,b}=b$. Each HNN extension embeds the previous group and is torsion-free by the result just proved. Let $G_{j+1}$ be the union after all pairs from $G_j$ have been treated. It remains countable because only countably many generators were added, and torsion-free because any element and any purported finite-order equation occur in a finite stage.

Repeat this whole process: new stable letters and their products must also be included in later pair lists. In the final increasing unionthe original $G$ embeds and torsion-freeness and countability persist. Any two nonidentity elements lie together in some $G_j$ and are conjugate in $G_{j+1}$. Thus every countable torsion-free group embeds in a countable torsion-free group in which all nonidentity elements are conjugate. If the initial group is nontrivial there are exactly two conjugacy classes; for the trivial group the stated condition is vacuous, or one can begin with $\mathbb{Z}$ instead. The listing is set-theoretic; no algorithm for recognizing nonidentity elements is assumed.