Isomers close to the doubly magic nucleus 78Ni (Z=28, N=50) provide essential information on the shell evolution and shape coexistence far from stability. The existence of a long-lived isomeric state in 76Cu has been debated for a long time. We have performed high-precision mass measurements of 76Cu with the JYFLTRAP double Penning trap mass spectrometer at the Ion Guide Isotope Separator On-Line facility and confirm the existence of such an isomeric state with an excitation energy Ex=64.8(25) keV. Based on the ratio of detected ground- and isomeric-state ions as a function of time, we show that the isomer is the shorter-living state previously considered as the ground state of 76Cu. The result can potentially change the conclusions made in previous works related to the spin-parity and charge radius of the 76Cu ground state. Additionally, the new 76Cu(n,γ) reaction Q-value has an impact on the astrophysical rapid neutron-capture process.