  Let $K/k$ be an abelian extension of number fields with a distinguished place
of $k$ that splits totally in $K$. In that situation, the abelian rank one
Stark conjecture predicts the existence of a unit in $K$, called the Stark
unit, constructed from the values of the $L$-functions attached to the
extension. In this paper, assuming the Stark unit exists, we prove index
formulae for it. In a second part, we study the solutions of the index formulae
and prove that they admit solutions unconditionally for quadratic, quartic and
sextic (with some additional conditions) cyclic extensions. As a result we
deduce a weak version of the conjecture ("up to absolute values") in these
cases and precise results on when the Stark unit, if it exists, is a square.
