  Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be
the ring of polynomials in a central variable over $H$, and let $H(T)$ be its
quotient skew field. We show that every intermediate division ring between $H$
and $H(T)$ is itself of the form $H(f)$, for some $f$ in the center of $H(T)$.
This generalizes the classical L\"uroth's theorem. More generally, we extend
Igusa's theorem characterizing the transcendence degree 1 subfields of rational
function fields, from fields to division rings.
