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Using combined data from the Relativistic Heavy Ion and Large Hadron Colliders, we constrain the shear and bulk viscosities of quark-gluon plasma (QGP) at temperatures of ∼ 150 – 350 MeV . We use Bayesian inference to translate experimental and theoretical uncertainties into probabilistic constraints for the viscosities. With Bayesian model averaging we propagate an estimate of the model uncertainty generated by the transition from hydrodynamics to hadron transport in the plasma's final evolution stage, providing the most reliable phenomenological constraints to date on the QGP viscosities.