  Aim: We present a high-spatial and high-spectral resolution observation of
the well-studied K giant Aldebaran with AMBER at the Very Large Telescope
Interferometer (VLTI). Our aim is to spatially resolve the outer atmosphere
(so-called MOLsphere) in individual CO first overtone lines and derive its
physical properties, which are important for understanding the mass-loss
mechanism in normal (i.e., non-Mira) K--M giants. Methods: Aldebaran was
observed between 2.28 and 2.31 micron with a projected baseline length of 10.4m
and a spectral resolution of 12000. Results: The uniform-disk diameter observed
in the CO first overtone lines is 20--35% larger than is measured in the
continuum. We have also detected a signature of inhomogeneities in the
CO-line-forming region on a spatial scale of ~45 mas, which is more than twice
as large as the angular diameter of the star itself. While the MARCS
photospheric model reproduces the observed spectrum well, the angular size in
the CO lines predicted by the MARCS model is significantly smaller than
observed. This is because the MARCS model with the parameters of Aldebaran has
a geometrical extension of only ~2% (with respect to the stellar radius). The
observed spectrum and interferometric data in the CO lines can be
simultaneously reproduced by placing an additional CO layer above the MARCS
photosphere. This CO layer is extended to 2.5 +/- 0.3 stellar radii with CO
column densities of 5x10^{19}--2x10^{20} cm^-2 and a temperature of 1500 +/-
200 K. Conclusions: The high spectral resolution of AMBER has enabled us to
spatially resolve the inhomogeneous, extended outer atmosphere (MOLsphere) in
the individual CO lines for the first time in a K giant. Our modeling of the
MOLsphere of Aldebaran suggests a rather small gradient in the temperature
distribution above the photosphere up to 2--3 stellar radii.
