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We provide the most precise measurement of the WW production cross section in pp collisions to date at a center of mass energy of 1.96 TeV, and set limits on the associated trilinear gauge couplings. The WW → ℓνℓ’ν(ℓ,ℓ’ = e, μ) decay channels are analyzed in 1 fb-1 of data collected by the D0 detector at the Fermilab Tevatron Collider. The measured cross section is σ(pp → WW) = 11.5 ± 2.1 (stat + syst) ± 0.7 (lumi) pb. One- and two-dimensional 95% C.L. limits on trilinear gauge couplings are provided.