Summary: This paper proves almost sure convergence for the self-attracting diffusion on the unit sphere \[ dX_t=\nu \circ dW_{t}(X_t)- a\int_{0}^{t}\nabla_{\mathbb{S}^n}V_{X_s}(X_t) dsdt, \;\;\;X_0=x\in \mathbb{S}^n, \] where \(\nu >0\), \(a < 0\), \(V_y(x)=\langle x,y\rangle \) is the usual scalar product on \(\mathbb{R}^{n+1}\), \(\circ\) stands for the Stratonovich differential and \((W_{t}(\cdot))_{t\geq 0}\) is a Brownian vector field on \(\mathbb{S}^n\). From this we deduce the almost sure convergence of the real-valued self-attracting diffusion \[ d\vartheta_{t}=\nu dW_{t}+a\int_{0}^{t}\sin (c(\vartheta_{t}- \vartheta_{s}))dsdt, \] where \((W_t)_{t\geq 0}\) is a real Brownian motion and \(c>0\).