  Asymptotic safety (an ultraviolet fixed point with finite-dimensional
critical surface) offers the possibility that a predictive theory of quantum
gravity can be obtained from the quantization of classical general relativity.
However, it is unclear what becomes of the singularities of classical general
relativity, which, it is hoped, might be resolved by quantum effects. We study
dust collapse with a running gravitational coupling and find that a future
singularity can be avoided if the coupling becomes exactly zero at some finite
energy scale. The singularity can also be avoided (pushed off to infinite
proper time) if the coupling approaches zero sufficiently rapidly at high
energies. However, the evolution deduced from perturbation theory still implies
a singularity at finite proper time.
