© 2015 American Physical Society. In recent models, the neutron-rich Ni isotopes around N=40 are predicted to exhibit multiple low-energy excited 0+ states attributed to neutron and proton excitations across both the N=40 and Z=28 shell gaps. In Ni68, the three observed 0+ states have been interpreted in terms of triple shape coexistence between spherical, oblate, and prolate deformed shapes. In the present work a new (02+) state at an energy of 1567 keV has been discovered in Ni70 by using β-delayed, γ-ray spectroscopy following the decay of Co70. The precipitous drop in the energy of the prolate-deformed 0+ level between Ni68 and Ni70 with the addition of two neutrons compares favorably with results of Monte Carlo shell-model calculations carried out in the large fpg9/2d5/2 model space, which predict a 02+ state at 1525 keV in Ni70. The result extends the shape-coexistence picture in the region to Ni70 and confirms the importance of the role of the tensor component of the monopole interaction in describing the structure of neutron-rich nuclei.