  We present K-band interferometric measurements of the limb-darkened intensity
profile of the M4 giant star psi Phoenicis obtained with VLTI/VINCI.
High-precision squared visibility amplitudes in the second lobe of the
visibility function were obtained employing two 8.2 m UTs. This succeeded one
month after light from UTs was first combined for interferometric fringes. In
addition, we sampled the visibility function at small spatial frequencies using
the 40cm test siderostats. Our measurement constrains the diameter of the star
as well as its CLV. We construct a spherical hydrostatic PHOENIX model
atmosphere based on spectrophotometric data from the literature and confront
its CLV prediction with our interferometric measurement. We compare as well CLV
predictions by plane-parallel hydrostatic PHOENIX, ATLAS9, and ATLAS12 models.
We find that the Rosseland angular diameter as predicted by comparison of the
spherical PHOENIX model with spectrophotometry is in good agreement with our
interferometric diameter measurement. The shape of our measured visibility
function in the second lobe is consistent with all considered PHOENIX and ATLAS
model predictions, and significantly different from UD and FDD models. We
derive high-precision fundamental parameters for psi Phe, namely a Rosseland
angular diameter of 8.13 +/- 0.2 mas, with the Hipparcos parallax corresponding
to a Rosseland linear radius R of 86 +/- 3 Rsun and an effective temperature of
3550 +/- 50 K, with R corresponding to a luminosity of log (L/Lsun)=3.02 +/-
0.06. Together with evolutionary models, these values are consistent with a
mass of 1.3 +/- 0.2 Msun, and a surface gravity of log g = 0.68 +/- 0.11.
