Neutron capture nucleosynthesis in massive stars plays an important role in galactic chemical evolution as well as for the analysis of abundance patterns in very old metal-poor halo stars. The so-called weak s-process component, which is responsible for most of the s abundances between Fe and Sr, turned out to be very sensitive to the stellar neutron capture cross sections in this mass region and, in particular, of isotopes near the seed distribution around Fe. Activation measurements in a quasistellar neutron spectrum corresponding to a thermal energy of kT = 25 keV have been carried out on 58 Fe, 59 Co, 64 Ni, 63 Cu, and 65 Cu. By a series of repeated irradiations with different experimental conditions, uncertainties between 3.0% and 4.6% could be achieved, factors of 2 to 3 more accurate than previous data. Compared to previous measurements, severe discrepancies were found for 63,65 Cu. The consequences of these results have been studied by detailed model calculations for convective core He burning and convective shell C burning in massive stars.