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Cross sections for the process e+e−→K0SK0SJ/ψ at center-of-mass energies from 4.128 to 4.950 GeV are measured using data samples with a total integrated luminosity of 21.2 fb−1 collected by the BESIII detector operating at the BEPCII storage ring. The Y(4230) state is observed in the energy dependence of the e+e−→K0SK0SJ/ψ cross section for the first time with a statistical significance of 26.0σ. In addition, an enhancement around 4.710 GeV, called the Y(4710), is seen with a statistical significance of 4.2σ. There is no clear structure around 4.484 GeV. Using a fit with a coherent sum of three Breit-Wigner functions, we determine the mass and width of the Y(4230) state to be 4226.9±6.6±21.9 MeV/c2 and 71.7±16.2±31.4 MeV, respectively, and the mass and width of the Y(4710) state to be 4704.0±52.3±69.5 MeV/c2 and 183.2±114.0±90.8 MeV, respectively, where the first uncertainties are statistical and the second are systematic. In addition, the average Born cross section ratio of e+e−→K0SK0SJ/ψ to e+e−→K+K−J/ψ is measured to be 0.388+0.035−0.028±0.016, or 0.426+0.038−0.031±0.018 if three-body phase space is considered.