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We present new measurements of radiative lifetimes for six energy levels of singly ionized uranium, U II, using laser-induced fluorescence technique. One of the levels, 5f³6d7p⁶M_13/2 at 26191cm^–1, decays by a transition at 3859.6Å. This line has recently been observed in the spectrum of the metal-poor star CS1082-001, the first detection of uranium outside the solar system. The λ3859 line can be used as the presently most accurate cosmochronometer (Cayrel et al., 2001Natur.409..691C). Our value of the lifetime of the ⁶M_13/2 level is 18.6±0.7 ns, and it confirms the f-value used in the Nature article by Cayrel et al. (2001Natur.409..691C), which is based on an experimental lifetime of 20±5ns (Chen & Borzileri, 1981, J. Chem. Phys. 74, 6063). The new measurement also removes the doubt about the choice between that value and other f-values in the literature, differing by a factor of 3. Adopting the same branching fraction as Chen & Borzileri (1981, J. Chem. Phys. 74, 6063) for the 3859.6Å line, we derive a gf-value of 0.68, which is 8% higher than the value used by Cayrel et al. (2001Natur.409..691C). Of significance for the chronometer is also the reduced uncertainty of the radiative lifetime, 4% compared to 25%, and consequently of the f-value, which should decrease the uncertainty in the determination of the stellar age considerably.