Self-attracting diffusions are processes satisfying an SDE of the form $$ X_t=B_t+\int_0^t\int_0^s f(X_s-X_u)\,duds $$ where $B$ is a one-dimensional Brownian motion and $f$ is an odd decreasing function such that $f\in{\scr C}^1({R})$. It is demonstrated that if there exist $\eta >0$, $\gamma\ge1$, and $C_\gamma>0$ such that $$ |f(x)-f(y)|\ge C_\gamma|x-y|^\gamma \, {\rm for all}\, |x-y|\le\eta, $$ then, for any $µ<1/(1+\gamma)$, $$ \lim_{t\to\infty}\left\{\left({{t}\over{\ln t}}\right)^µ\sup_{s\ge t} |X_s-X_t|\right\}=0 {\rm a.s.} $$