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We report on high-resolution echelle spectroscopy of 20 giant stars in the Galactic old open cluster NGC 6791, obtained with Hydra at the WIYN telescope. High-precision radial velocity allows us to isolate 15 bona fide cluster members. From 10 of them we derive a global [M/H]=+0.39±0.05. We therefore confirm that NGC 6791 is extremely metal-rich, exhibits a few marginally subsolar abundance ratios, and within the resolution of our spectra does not show evidence of spread in metal abundance. With these new data we rederive the fundamental cluster parameters, suggesting that it is about 8 Gyr old and 4.3 kpc from the Sun. The combination of its chemical properties, age, position, and Galactic orbit hardly makes NGC 6791 a genuine Population I open cluster. We discuss possible interpretations of the cluster peculiarities, suggesting that the cluster might be what remains of a much larger system whose initial potential well could have been sufficient to produce high-metallicity stars and which has been depopulated by the tidal field of the Galaxy. Alternatively, its current properties may be explained by the perturbation of the Galactic bar on an object that originated well inside the solar ring, where the metal enrichment was very fast.