We report laser-induced-fluorescence spectroscopy of the ${}^{1}S_{0}\rightarrow{}^{3}P_{1}$ intercombination transition in neutral zinc at $307.6~\mathrm{nm}$. Isotope shifts are measured for all stable isotopes with kHz-level precision, improving previous data by about two orders of magnitude. For $^{67}\mathrm{Zn}$, we resolve the excited-state hyperfine structure and determine $\delta\nu_{\mathrm{COG}}^{67,64}=1085.933(4)~\mathrm{MHz}$, $A=608.922(1)~\mathrm{MHz}$, and $B=-18.995(4)~\mathrm{MHz}$. A King-plot comparison with the ${}^{1}S_{0}\rightarrow{}^{1}P_{1}$ transition at $214~\mathrm{nm}$ gives field- and mass-shift parameters of $F_{307.6,214}=1.17(5)$ and $K=-153(60)~\mathrm{GHz\,u}$. These results provide the spectroscopic basis for narrow-line cooling and precision measurements based on zinc, including the development of an optical clock.