The neutron capture cross section of Bi-209 has been measured at the CERN n_TOF facility by employing the pulse-height-weighting technique. Improvements over previous measurements are mainly because of an optimized detection system, which led to a practically negligible neutron sensitivity. Additional experimental sources of systematic error, such as the electronic threshold in the detectors, summing of gamma-rays, internal electron conversion, and the isomeric state in bismuth, have been taken into account. gamma-Ray absorption effects inside the sample have been corrected by employing a nonpolynomial weighting function. Because Bi-209 is the last stable isotope in the reaction path of the stellar s-process, the Maxwellian averaged capture cross section is important for the recycling of the reaction flow by alpha decays. In the relevant stellar range of thermal energies between kT=5 and 8 keV our new capture rate is about 16% higher than the presently accepted value used for nucleosynthesis calculations. At this low temperature an important part of the heavy Pb-Bi isotopes are supposed to be synthesized by the s-process in the He shells of low mass, thermally pulsing asymptotic giant branch stars. With the improved set of cross sections we obtain an s-process fraction of 19 +/- 3% of the solar bismuth abundance, resulting in an r-process residual of 81 +/- 3%. The present (n,gamma) cross-section measurement is also of relevance for the design of accelerator driven systems based on a liquid metal Pb/Bi spallation target< Réduire