  Let $P$ be a set of $n$ points in strictly convex position in the plane. Let
$D_n$ be the graph whose vertex set is the set of all line segments with
endpoints in $P$, where disjoint segments are adjacent. The chromatic number of
this graph was first studied by Araujo, Dumitrescu, Hurtado, Noy, and Urrutia
[2005] and then by Dujmovi\'c and Wood [2007]. Improving on their estimates, we
prove the following exact formula: $$\chi(D_n) = n - \left\lfloor \sqrt{2n +
\tfrac{1}{4}} - \tfrac{1}{2}\right\rfloor.$$
