  Observations of exotic structures in the $J/\psi p$ channel, that we refer to
as pentaquark-charmonium states, in $\Lambda_b^0\to J/\psi K^- p$ decays are
presented. The data sample corresponds to an integrated luminosity of 3/fb
acquired with the LHCb detector from 7 and 8 TeV pp collisions. An amplitude
analysis is performed on the three-body final-state that reproduces the
two-body mass and angular distributions. To obtain a satisfactory fit of the
structures seen in the $J/\psi p$ mass spectrum, it is necessary to include two
Breit-Wigner amplitudes that each describe a resonant state. The significance
of each of these resonances is more than 9 standard deviations. One has a mass
of $4380\pm 8\pm 29$ MeV and a width of $205\pm 18\pm 86$ MeV, while the second
is narrower, with a mass of $4449.8\pm 1.7\pm 2.5$ MeV and a width of $39\pm
5\pm 19$ MeV. The preferred $J^P$ assignments are of opposite parity, with one
state having spin 3/2 and the other 5/2.
