  Generating quantum entanglement in large systems on time scales much shorter
than the coherence time is key to powerful quantum simulation and computation.
Trapped ions are among the most accurately controlled and best isolated quantum
systems with low-error entanglement gates operated via the vibrational motion
of a few-ion crystal within tens of microseconds. To exceed the level of
complexity tractable by classical computers the main challenge is to realise
fast entanglement operations in large ion crystals. The strong dipole-dipole
interactions in polar molecule and Rydberg atom systems allow much faster
entangling gates, yet stable state-independent confinement comparable with
trapped ions needs to be demonstrated in these systems. Here, we combine the
benefits of these approaches: we report a $700\,\mathrm{ns}$ two-ion entangling
gate which utilises the strong dipolar interaction between trapped Rydberg ions
and produce a Bell state with $78\%$ fidelity. The sources of gate error are
identified and a total error below $0.2\%$ is predicted for
experimentally-achievable parameters. Furthermore, we predict that residual
coupling to motional modes contributes $\sim 10^{-4}$ gate error in a large ion
crystal of 100 ions. This provides a new avenue to significantly speed up and
scale up trapped ion quantum computers and simulators.
