In CDF we have observed the reactions p+p→p+X+p, with X being a centrally produced J/ψ, ψ(2S), or χc0, and γγ→μ+μ− in pp¯ collisions at s√=1.96  TeV. The event signature requires two oppositely charged central muons, and either no other particles or one additional photon detected. Exclusive vector meson production is as expected for elastic photoproduction, γ+p→J/ψ(ψ(2S))+p, observed here for the first time in hadron-hadron collisions. We also observe exclusive χc0→J/ψ+γ. The cross sections dσdy|y=0 for J/ψ, ψ(2S), and χc0 are 3.92±0.25(stat)±0.52(syst)  nb, 0.53±0.09(stat)±0.10(syst)  nb, and 76±10(stat)±10(syst)  nb, respectively, and the continuum is consistent with QED. We put an upper limit on the cross section for Odderon exchange in exclusive J/ψ production.