  We report on the measurement of the $\gamma p \rightarrow J/\psi p$ cross
section from $E_\gamma = 11.8$ GeV down to the threshold at $8.2$ GeV using a
tagged photon beam with the GlueX experiment. We find the total cross section
falls toward the threshold less steeply than expected from two-gluon exchange
models. The differential cross section $d\sigma /dt$ has an exponential slope
of $1.67 \pm 0.39$ GeV$^{-2}$ at $10.7$ GeV average energy. The LHCb pentaquark
candidates $P_c^+$ can be produced in the $s$-channel of this reaction. We see
no evidence for them and set model-dependent upper limits on their branching
fractions $\mathcal{B}(P_c^+ \rightarrow J/\psi p)$ and cross sections
$\sigma(\gamma p \to P_c^+)\times\mathcal{B}(P_c^+ \to J/\psi p) $.
