When an accretion disc falls prey to the runaway instability, a large portion of its mass is devoured by the black hole within a few dynamical times. Despite decades of effort, it is still unclear under what conditions such an instability can occur. The technically most advanced relativistic simulations to date were unable to find a clear sign for the onset of the instability. In this work, we present three-dimensional relativistic hydrodynamics simulations of accretion discs around black holes in dynamical space–time. We focus on the configurations that are expected to be particularly prone to the development of this instability. We demonstrate, for the first time, that the fully self-consistent general relativistic evolution does indeed produce a runaway instability.