Let ( ${𝕊}^1,d_{{𝕊}^1}$ S 1 , d S 1 ) be the unit circle in ℝ2 endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure μto admit a well defined Fréchet mean on ( ${𝕊}^1,d_{{𝕊}^1}$ S 1 , d S 1 ). We derive a new sufficient condition of existenceP(α, ϕ) with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.