  We present optical long-slit spectra of the Virgo dwarf elliptical galaxy VCC
510 at high spectral and spatial resolution. Heliocentric velocities and
velocity dispersions as functions of galaxy radius are derived by deconvolving
line-of-sight velocity distributions. A maximum rotation v_rot=8 km/s inside
half the effective radius (re~20 arcsec) and a mean, radially flat velocity
dispersion sigma=44 km/s are measured. The core extending over the inner 2
arcsec (~140 pc) is found to rotate in the opposite sense with v_rot(core) ~
-1/2 v_rot. VCC 510 (M_B ~ -15.7) is therefore by far the faintest and smallest
galaxy with a counter-rotating core known. From the main body rotation and the
velocity dispersion profile we deduce that VCC 510 is anisotropic and clearly
not entirely supported by rotation. By comparison of Lick absorption-line
indices with stellar population models we derive an old luminosity-weighted age
(10 Gyr) and sub-solar metallicity ([Z/H]=-0.4) inside the effective radius.
There is tentative evidence that the counter-rotating core might be younger and
less alpha/Fe enhanced. From the stellar population parameters we obtain a
total stellar mass-to-light ratio of ~3.6 (M/L_B)_sun which is significantly
lower than a rough dynamical estimate obtained from the kinematics through the
virial theorem (~15). This discrepancy hints toward the possible presence of
dark matter in the centre of VCC 510. We discuss the origin of the
counter-rotating core and exclude fly-by encounters as a viable possibility.
Gas accretion or galaxy merging provide more likely explanations. VCC 510 is
therefore the direct observational evidence that such processes do occur in
cluster satellite galaxies on dwarf galaxy scales.
