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We present a high-resolution (R=75000, S/N∼500) spectroscopic analysis of the bright (V=11.7), extreme halo giant CS 31082-001 ([Fe/H]=-2.9), obtained in an ESO-VLT Large Programme dedicated to very metal-poor stars. We find CS 31082-001 to be extremely rich in r-process elements, comparable in this respect only to the similarly metal-poor, but carbon-enriched, giant CS 22892-052. As a result of the extreme overabundance of the heaviest r-process elements, and negligible blending from CH and CN molecular lines, a reliable measurement is obtained of the U II line at 386nm, for the first time in a halo star, along with numerous lines of Th II, as well as lines of 25 other r-process elements. Abundance estimates for a total of 43 elements (44 counting Hydrogen) are reported in CS 31082-001, almost half of the entire periodic table. The main atmospheric parameters of CS 31082-001 are as follows: T_eff=4825±50K, logg=1.5±0.3 (cgs), [Fe/H]=-2.9±0.1 (in LTE), and microturbulence 1.8±0.2km/s. Carbon and nitrogen are not significantly enhanced relative to iron. As usual in giant stars, Li is depleted by dilution (log(Li/H)=0.85). The α-elements show the usual enhancements with respect to iron, with [O/Fe]=0.6±0.2 (from [O I] 6300Å), [Mg/Fe]=0.45±0.16, [Si/Fe]=0.24±0.1, and [Ca/Fe]=0.41±0.08, while [Al/Fe] is near -0.5. The r-process elements show unusual patterns: among the lightest elements (Z∼40), Sr and Zr follow the Solar r-element distribution, but Ag is down by 0.8dex. All elements with 56≤Z≤72 follow the Solar r-element pattern, reduced by about 1.25dex. Accordingly, the [r/Fe] enhancement is about +1.7dex (a factor of 50), very similar to that of CS 22892-052. Pb, in contrast, seems to be below the shifted Solar r-process distribution, possibly indicating an error in the latter, while thorium is more enhanced than the lighter nuclides. In CS 31082-001, log(Th/Eu) is -0.22±0.07, higher than in the Solar System (-0.46) or in CS 22892-052 (-0.66). If CS 31082-001 and CS 22892-052 have similar ages, as expected for two extreme halo stars, this implies that the production ratios were different by about 0.4dex for the two objects. Conversely, if the Th/Eu production ratio were universal, an age of 15Gyr for CS 22892-052 would imply a negative age for CS 31082-001. Thus, while a universal production ratio for the r-process elements seems to hold in the interval 56≤Z≤72, it breaks down in the actinide region. When available, the U/Th is thus preferable to Th/Eu for radioactive dating, for two reasons: (i) because of its faster decay rate and smaller sensitivity to observational errors, and (ii) because the inital production ratio of the neighboring nuclides ²³⁸U and ²³²Th is more robustly predicted than the ¹⁵¹Eu/²³²Th ratio. Our current best estimate for the age of CS 31082-001 is 14.0±2.4Gyr. However, the computed actinide production ratios should be verified by observations of daughter elements such as Pb and Bi in the same star, which are independent of the subsequent history of star formation and nucelosynthesis in the Galaxy.