  We measure the forward--backward asymmetry of the production of top quark and
antiquark pairs in proton-antiproton collisions at center-of-mass energy
$\sqrt{s} = 1.96~\mathrm{TeV}$ using the full data set collected by the
Collider Detector at Fermilab (CDF) in Tevatron Run II corresponding to an
integrated luminosity of $9.1~\rm{fb}^{-1}$. 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 $A_{\text{FB}}^{t\bar{t}} = 0.12 \pm 0.13$, 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 $A_{\text{FB}}^{t\bar{t}}$ in both final states yields
$A_{\text{FB}}^{t\bar{t}}=0.160\pm0.045$, which is consistent with the SM
predictions. We also measure the differential asymmetry as a function of
$\Delta y$. A linear fit to $A_{\text{FB}}^{t\bar{t}}(|\Delta y|)$, assuming
zero asymmetry at $\Delta y=0$, yields a slope of $\alpha=0.14\pm0.15$,
consistent with the SM prediction and the previous CDF determination in the
final state with a single charged lepton. The combined slope of
$A_{\text{FB}}^{t\bar{t}}(|\Delta y|)$ in the two final states is
$\alpha=0.227\pm0.057$, which is $2.0\sigma$ larger than the SM prediction.
