Countable torsion-free embedding with two conjugacy classes

Every nontrivial countable torsion-free group embeds in a countable torsion-free group with precisely two conjugacy classes, the identity and all nonidentity elements. At each stage, enumerate all ordered pairs of nonidentity elements of the current group and successively adjoin HNN extensions identifying their infinite cyclic subgroups. Britton's lemma preserves embeddings and torsion in an HNN extension preserves torsion-freeness. Repeat on the resulting countable union, so pairs involving newly introduced elements are eventually included. A final increasing union is countable and torsion-free, and any two nonidentity elements become conjugate at a later stage. No effective enumeration of nonidentity elements is claimed.