  Let $(\S^1,d_{\S^1})$ be the unit circle in $\R^2$ endowed with the arclength
distance. We give a sufficient and necessary condition for a general
probability measure $\mu$ to admit a well defined Fr\'echet mean on
$(\S^1,d_{\S^1})$. %This criterion allows to recover already known sufficient
conditions of existence. We derive a new sufficient condition of existence
$P(\alpha,\varphi)$ with no restriction on the support of the measure. Then, we
study the convergence of the empirical Fr\'echet mean to the Fr\'echet mean and
we give an algorithm to compute it.
