  We report the observations in the K band of the red supergiant star alpha
Orionis and of the bright giant star alpha Herculis with the FLUOR beamcombiner
at the IOTA interferometer. The high quality of the data allows us to estimate
limb-darkening and derive precise diameters in the K band which combined with
bolometric fluxes yield effective temperatures. In the case of Betelgeuse, data
collected at high spatial frequency although sparse are compatible with
circular symmetry and there is no clear evidence for departure from circular
symmetry. We have combined the K band data with interferometric measurements in
the L band and at 11.15 micron. The full set of data can be explained if a 2055
K layer with optical depths $\tau_{K}=0.060\pm0.003$, $\tau_{L}=0.026\pm0.002$
and $\tau_{11.15\mu m}=2.33\pm0.23$ is added 0.33 $R_{\star}$ above the
photosphere providing a first consistent view of the star in this range of
wavelengths. This layer provides a consistent explanation for at least three
otherwise puzzling observations: the wavelength variation of apparent diameter,
the dramatic difference in limb darkening between the two supergiant stars, and
the previously noted reduced effective temperature of supergiants with respect
to giants of the same spectral type. Each of these may be simply understood as
an artifact due to not accounting for the presence of the upper layer in the
data analysis. This consistent picture can be considered strong support for the
presence of a sphere of warm water vapor, proposed by Tsuji (2000) when
interpreting the spectra of strong molecular lines.
