We use measurements of the peak photon energy and bolometric fluence of 119 gamma-ray bursts (GRBs) extending over the redshift range of $0.3399 leq z leq 8.2$ to simultaneously determine cosmological and Amati relation parameters in six different cosmological models. The resulting Amati relation parameters are almost identical in all six cosmological models, thus validating the use of the Amati relation in standardizing these GRBs. The GRB data cosmological parameter constraints are consistent with, but significantly less restrictive than, those obtained from a joint analysis of baryon acoustic oscillation and Hubble parameter measurements.