In the nuclear mass range A≈60 to 90 of the solar abundance distribution the weak s-process component is the dominant contributor. In this scenario, which is related to massive stars, the overall neutron exposure is not sufficient for the s process to reach mass flow equilibrium. Hence, abundances and isotopic ratios are very sensitive to the neutron capture cross sections of single isotopes, and nucleosynthesis models need accurate experimental data. In this work we report on a new measurement of the Cu63(n,γ) cross section for which the existing experimental data show large discrepancies. The Cu63(n,γ) cross section at kBT=25 keV was determined via activation with a quasistellar neutron spectrum at the Joint Research Centre (JRC) in Geel, and the energy dependence was determined with the time-of-flight technique and the calorimetric 4πBaF2 detector array DANCE at the Los Alamos National Laboratory. We provide new cross section data for the whole astrophysically relevant energy range.