  Aims: We present the first in-depth study of the kinematic properties and
derive the structural parameters of NGC 4372 based on the fit of a Plummer
profile and a rotating, physical model. We explore the link between internal
rotation to different cluster properties and together with similar studies of
more GCs, we put these in the context of globular cluster formation and
evolution. Methods: We present radial velocities for 131 cluster member stars
measured from high-resolution FLAMES/GIRAFFE observations. Their membership to
the GC is additionally confirmed from precise metallicity estimates. Using this
kinematic data set we build a velocity dispersion profile and a systemic
rotation curve. Additionally, we obtain an elliptical number density profile of
NGC 4372 based on optical images using a MCMC fitting algorithm. From this we
derive the cluster's half-light radius and ellipticity as r_h=3.4'+/-0.04' and
e=0.08+/-0.01. Finally, we give a physical interpretation of the observed
morphological and kinematic properties of this GC by fitting an axisymmetric,
differentially rotating, dynamical model. Results: Our results show that NGC
4372 has an unusually high ratio of rotation amplitude to velocity dispersion
(1.2 vs. 4.5 km/s) for its metallicity. This, however, puts it in line with two
other exceptional, very metal-poor GCs - M 15 and NGC 4590. We also find a mild
flattening of NGC 4372 in the direction of its rotation. Given its old age,
this suggests that the flattening is indeed caused by the systemic rotation
rather than tidal interactions with the Galaxy. Additionally, we estimate the
dynamical mass of the GC M_dyn=2.0+/-0.5 x 10^5 M_Sun based on the dynamical
model, which constrains the mass-to-light ratio of NGC 4372 between 1.4 and 2.3
M_Sun/L_Sun, representative of an old, purely stellar population.

Astrophysics of Galaxies