A narrow pentaquark state, P-c(4312)(+), decaying to J/psi p, is discovered with a statistical significance of 7.3 sigma in a data sample of Lambda(0)(b) -> J/psi pK(-) decays, which is an order of magnitude larger than that previously analyzed by the LHCb Collaboration. The P-c(4450)(+) pentaquark structure formerly reported by LHCb is confirmed and observed to consist of two narrow overlapping peaks, P-c(4440)(+) and P-c(4457)(+), where the statistical significance of this two-peak interpretation is 5.4 sigma. The proximity of the Sigma(+)(c)(D) over bar (0) and Sigma(+)(c)(D) over bar (*0) thresholds to the observed narrow peaks suggests that they play an important role in the dynamics of these states.