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We present a study of μμμμ, eeee, and μμee events using 1 fb-1 of data collected with the D0 detector at the Fermilab Tevatron pp Collider at √s = 1.96 TeV. Requiring the lepton pair masses to be greater than 30 GeV, we observe one event, consistent with the expected background of 0.13 ± 0.03 events and with the predicted standard model ZZ and Zγ* production of 1.71 ± 0.15 events. We set an upper limit on the ZZ and Zγ* cross section of 4.4 pb at the 95% C.L. We also derive limits on anomalous neutral trilinear ZZZ and ZZγ* gauge couplings. The one-parameter 95% C.L. coupling limits with a form-factor scale Λ = 1.2 TeV are -0.28< ƒ40Z < 0.28, -0.31 < ƒ50z < 0.29, -0.26 < ƒ40γ < 0.26, and -0.30 < ƒ50γ < 0.28.