  In this paper we present high resolution VLT/FLAMES observations of red giant
stars in the massive intermediate-age Large Magellanic Cloud star cluster NGC
1846, which, on the basis of its extended main-sequence turn-off (EMSTO),
possesses an internal age spread of ~300 Myr. We describe in detail our target
selection and data reduction procedures, and construct a sample of 21 stars
possessing radial velocities indicating their membership of NGC 1846 at high
confidence. We consider high-resolution spectra of the planetary nebula Mo-17,
and conclude that this object is also a member of the cluster. Our measured
radial velocities allow us to conduct a detailed investigation of the internal
kinematics of NGC 1846, the first time this has been done for an EMSTO system.
The key result of this work is that the cluster exhibits a significant degree
of systemic rotation, of a magnitude comparable to the mean velocity
dispersion. Using an extensive suite of Monte Carlo models we demonstrate that,
despite our relatively small sample size and the substantial fraction of
unresolved binary stars in the cluster, the rotation signal we detect is very
likely to be genuine. Our observations are in qualitative agreement with the
predictions of simulations modeling the formation of multiple populations of
stars in globular clusters, where a dynamically cold, rapidly rotating second
generation is a common feature. NGC 1846 is less than one relaxation time old,
so any dynamical signatures encoded during its formation ought to remain
present.
