A partial wave analysis of the ppˉ mass-threshold enhancement in the reaction J/ψ→γppˉ is used to determine: its JPC quantum numbers to be 0−+; its peak mass to be below threshold at M=1832−5+19(stat.)−17+18(syst.)±19(model)MeV/c2; and its total width to be Γ<76MeV/c2 at the 90% C.L. The product branching ratio is measured to be B(J/ψ→γX(ppˉ))B(X(ppˉ)→ppˉ)=(9.0−1.1+0.4(stat.)−5.0+1.5(syst.)±2.3(model))×10−5. A similar analysis performed on ψ′→γppˉ decays shows, for the first time, the presence of a corresponding enhancement with a production rate relative to that for J/ψ decays of R=(5.08−0.45+0.71(stat.)−3.58+0.67(syst.)±0.12(model)).