Observations of exotic structures in the J/ψp channel, which we refer to as charmonium-pentaquark states, in Λ0b→J/ψK−p decays are presented. The data sample corresponds to an integrated luminosity of 3 fb−1 acquired with the LHCb detector from 7 and 8 TeV pp collisions. An amplitude analysis of the three-body final state reproduces the two-body mass and angular distributions. To obtain a satisfactory fit of the structures seen in the J/ψp 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±8±29 MeV and a width of 205±18±86 MeV, while the second is narrower, with a mass of 4449.8±1.7±2.5 MeV and a width of 39±5±19 MeV. The preferred JP assignments are of opposite parity, with one state having spin 3/2 and the other 5/2.2 MoreReceived 13 July 2015DOI:https://doi.org/10.1103/PhysRevLett.115.072001This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.© 2015 CERN, for the LHCb Collaboration