  We report a measurement of exclusive $J/\psi$ photoproduction in Au$+$Au
ultra-peripheral collisions at $\sqrt{s_{_\mathrm{NN}}}=200$ GeV using the STAR
detector. For the first time, i) the rapidity correlation between $J/\psi$
photoproduction and neutron emission from nuclear breakups has been
experimentally measured; ii) nuclear suppression factors are measured for both
the coherent and incoherent $J/\psi$ production. At photon-nucleon
center-of-mass energy of 25.0 GeV, the coherent and incoherent $J/\psi$ cross
sections of Au nuclei are found to be $71\pm10\%$ and $36\pm7\%$, respectively,
of that of free protons. The stronger suppression observed in the incoherent
production provides a new experimental handle to study the initial-state parton
density in heavy nuclei. Data are compared with theoretical models
quantitatively.
