The radiative capture cross section of a highly pure (99.999%), 6.125(2) grams and 9.56(5)E-4 atoms/barn areal density 238U sample has been measured with the Total Absorption Calorimeter (TAC) in the 185 m flight path at the CERN neutron time-of-flight facility n_TOF. This measurement is in response to the NEA High Priority Request list, which demands an accuracy in this cross section of less than 3% below 25 keV. These data have undergone careful background subtraction, with special care being given to the background originating from neutrons scattered by the 238U sample. Pileup and dead-time effects have been corrected for. The measured cross section covers an energy range between 0.2 eV and 80 keV, with an accuracy that varies with neutron energy, being better than 4% below 25 keV and reaching at most 6% at higher energies.