  We complete the solution of the relative class number one problem for
function fields of curves over finite fields. Using work from two earlier
papers, this reduces to finding all function fields of genus 6 or 7 over
$\mathbb{F}_2$ with one of 40 prescribed Weil polynomials; one may then verify
directly that three of these fields admit an everywhere unramified quadratic
extension with trivial relative class group. The search is carried out by
carefully enumerating curves based on the Brill--Noether stratification of the
moduli spaces of curves in these genera, and particularly Mukai's descriptions
of the open strata.
