Collinear laser spectroscopy is performed on the 79Zn isotope at ISOLDE-CERN. The existence of a long-lived isomer with a few hundred milliseconds half-life is confirmed, and the nuclear spins and moments of the ground and isomeric states in Zn79 as well as the isomer shift are measured. From the observed hyperfine structures, spins I=9/2 and I=1/2 are firmly assigned to the ground and isomeric states. The magnetic moment μ (Zn79)=−1.1866(10)μN, confirms the spin-parity 9/2+ with a νg−19/2 shell-model configuration, in excellent agreement with the prediction from large scale shell-model theories. The magnetic moment μ (Zn79m)=−1.0180(12)μN supports a positive parity for the isomer, with a wave function dominated by a 2h−1p neutron excitation across the N=50 shell gap. The large isomer shift reveals an increase of the intruder isomer mean square charge radius with respect to that of the ground state, δ⟨r2c⟩79,79m=+0.204(6) fm2, providing first evidence of shape coexistence.