  By measuring second occurring times of factors of an infinite word $x$,
Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent of
repetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x)
\leq r_{\max} = \sqrt{10} - 3/2$ if $x$ is a Sturmian words. In this paper, we
determine the value $r_1$ such that there is no Sturmian word $x$ satisfying
$r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulation point of the set
of ${\rm rep}(x)$ when $x$ runs over the Sturmian words.
