A high-precision measurement of the branching fraction of the decay J/ψ → pp<over bar>η  is performed using (10 087 ± 44) x 10⁶ J/ψ events recorded by the BESIII detector at the BEPCII storage ring. The branching fractions of the two decays J/ψ → pp<over bar>η(η → γγ) and J/ψ → pp<over bar>η(η → π⁺π^-π⁰) are measured individually to be B(J/ψ → pp<over bar>η(η → γγ)) = (1.480 ± 0.001 ± 0.024) x 10^-3 and B(J/ψ → pp<over bar>η(η → π⁺π^-π⁰) = (1.557 ± 0.003 ± 0.038) x 10^-3, where the first uncertainties are statistical and the second systematic. Both results are compatible within their uncorrelated systematic uncertainties. The combined result is B(J/ψ → pp<over bar>η) = (1.495 ± 0.001 ± 0.023) x 10^-3, where the first uncertainty is the combined statistical uncertainty and the second one the combined systematic uncertainty of both analyses, incorporating correlations between them. In addition, the pp<over bar> threshold region is investigated for a potential threshold enhancement, and no evidence for one is observed.