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