Background: The prompt gamma-ray spectrum from fission is important for understanding the physics of nuclear fission, and also in applications involving fission. Relatively few measurements of the prompt gamma spectrum from Pu-239(n, f) have been published. Purpose: This experiment measured the multiplicity, individual gamma energy spectrum, and total gamma energy spectrum of prompt fission gamma rays from Pu-239(n, f) in the neutron energy range from thermal to 30 keV, to test models of fission and to provide information for applications. Method: Gamma rays from neutron-induced fission of Pu-239 were measured using the DANCE gamma-ray calorimeter. Fission events were tagged by detecting fission products in a parallel-plate avalanche counter in the center of DANCE. The measurements were corrected for detector response using a GEANT4 model of DANCE. A detailed analysis for the gamma rays from the 1(+) resonance complex at 10.93 eV is presented. Results: A six-parameter analytical parametrization of the fission gamma-ray spectrum was obtained. A Monte Carlo Hauser-Feshbach calculation provided good general agreement with the data, but some differences remain to be resolved. Conclusions: An analytic parametrization can be made of the gamma-ray multiplicity, energy distribution, and total-energy distribution for the prompt gamma rays following neutron-induced fission of Pu-239. This parametrization may be useful for applications. Modern Monte Carlo Hauser-Feshbach calculations can do a good job of calculating the fission gamma-ray emission spectrum, although some details remain to be understood. DOI: 10.1103/PhysRevC.87.044607