  We derive the asymptotic distribution of the total length $L_n$ of a
$\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$,
starting from $n$ individuals. There are two regimes: If
$\alpha\le1/2(1+\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit
distribution of index $\alpha$. Otherwise $L_n$ just has to be shifted by a
constant (depending on $n$) to get convergence to a nondegenerate limit
distribution. As a consequence, we obtain the limit distribution of the number
$S_n$ of segregation sites. These are points (mutations), which are placed on
the tree's branches according to a Poisson point process with constant rate.
