We measure the forward-backward asymmetry of the production of top-quark and antiquark pairs in proton-antiproton collisions at center-of-mass energy root s = 1.96 TeV using the full data set collected by the Collider Detector at Fermilab (CDF) in Tevatron Run II corresponding to an integrated luminosity of 9.1 fb(-1). The asymmetry is characterized by the rapidity difference between top quarks and antiquarks (Delta y) and measured in the final state with two charged leptons (electrons and muons). The inclusive asymmetry, corrected to the entire phase space at parton level, is measured to be A(FB)(t (t) over bar) = 0.12 +/- 0.13, consistent with the expectations from the standard model (SM) and previous CDF results in the final state with a single charged lepton. The combination of the CDF measurements of the inclusive A(FB)(t (t) over bar) in both final states yields A(FB)(t (t) over bar) 0.160 +/- 0.045, which is consistent with the SM predictions. We also measure the differential asymmetry as a function of Delta y. A linear fit to A(FB)(t (t) over bar) (vertical bar Delta y vertical bar),assuming zero asymmetry at Delta y = 0, yields a slope of alpha = 0.14 +/- 0.15, consistent with the SM prediction and the previous CDF determination in the final state with a single charged lepton. The combined slope of A(FB)(t (t) over bar) (vertical bar Delta y vertical bar) the two final states is alpha = 0.227 +/- 0.057, which is 2.0 sigma larger than the SM prediction.