Our goal is to demonstrate the potential of the interferometric AMBER
instrument linked with the Very Large Telescope Interferometer (VLTI)
fringe-tracking facility FINITO to derive high-precision stellar diameters. We
use commissioning data obtained on the bright single star V3879 Sgr. Locking
the interferometric fringes with FINITO allows us to record very low contrast
fringes on the AMBER camera. By fitting the amplitude of these fringes, we
measure the diameter of the target in three directions simultaneously with an
accuracy of 25 micro-arcseconds. We showed that V3879 Sgr has a round
photosphere down to a sub-percent level. We quickly reached this level of
accuracy because the technique used is independent from absolute calibration
(at least for baselines that fully span the visibility null). We briefly
discuss the potential biases found at this level of precision. The proposed
AMBER+FINITO instrumental setup opens several perspectives for the VLTI in the
field of stellar astrophysics, like measuring with high accuracy the oblateness
of fast rotating stars or detecting atmospheric starspots