The cross section of the Ni62(n,γ) reaction was measured with the time-of-flight technique at the neutron time-of-flight facility n-TOF at CERN. Capture kernels of 42 resonances were analyzed up to 200 keV neutron energy and Maxwellian averaged cross sections (MACS) from kT= 5-100 keV were calculated. With a total uncertainty of 4.5%, the stellar cross section is in excellent agreement with the the KADoNiS compilation at kT=30 keV, while being systematically lower up to a factor of 1.6 at higher stellar temperatures. The cross section of the Ni63(n,γ) reaction was measured for the first time at n-TOF. We determined unresolved cross sections from 10 to 270 keV with a systematic uncertainty of 17%. These results provide fundamental constraints on s-process production of heavier species, especially the production of Cu in massive stars, which serve as the dominant source of Cu in the solar system. © 2014 American Physical Society.