The masses of the neutron-rich copper isotopes Cu75-79 are determined using the precision mass spectrometer ISOLTRAP at the CERN-ISOLDE facility. The trend from the new data differs significantly from that of previous results, offering a first accurate view of the mass surface adjacent to the Z=28, N=50 nuclide Ni78 and supporting a doubly magic character. The new masses compare very well with large-scale shell-model calculations that predict shape coexistence in a doubly magic Ni78 and a new island of inversion for Z<28. A coherent picture of this important exotic region begins to emerge where excitations across Z=28 and N=50 form a delicate equilibrium with a spherical mean field.