We report a measurement of exclusive J/ψ and ψ(2s) photoproduction in Au+Au ultraperipheral collisions at sqrt[s_{NN}]=200 GeV using the STAR detector. For the first time, (i) the ψ(2s) photoproduction in midrapidity at the Relativistic Heavy-Ion Collider has been experimentally measured; (ii) nuclear suppression factors are measured for both the coherent and incoherent J/ψ production. At average photon-nucleon center-of-mass energy of 25.0 GeV, the coherent and incoherent J/ψ cross sections …