We report on the first low-energy Coulomb excitation measurements with radioactive Ipi=6- beams of odd-odd nuclei 68,70Cu. The beams were produced at ISOLDE, CERN and were post-accelerated by REX-ISOLDE to 2.83 MeV/nucleon. Gamma rays were detected with the MINIBALL spectrometer. The 6- beam was used to study the multiplet of states (3-, 4-, 5-, 6-) arising from the pi2p3/2 nu 1g9/2 configuration. The 4- state of the multiplet was populated via Coulomb excitation and the B(E2;6--->4-) value was determined in both nuclei. The results obtained illustrate the fragile stability of the Z=28 shell and N=40 subshell closures. A comparison with large-scale shell-model calculations using the 56Ni core shows the importance of the proton excitations across the Z=28 shell gap to the understanding of the nuclear structure in the neutron-rich nuclei with N approximately 40.