  A cone singularity is a normal affine variety $X$ with an effective
one-dimensional torus action with a unique fixed point $x\in X$ which lies in
the closure of any orbit of the $k^*$-action. In this article, we prove a
boundedness theorem for cone singularities in terms of their dimension,
singularities, and isotropies. Given $d$ and $N$ two positive integers and
$\epsilon$ a positive real number, we prove that the class of $d$-dimensional
$\epsilon$-log canonical cone singularities with isotropies bounded by $N$
forms a bounded family.
