We present a measurement of the top-quark mass in events containing two leptons (electrons or muons) with a large transverse momentum, two or more energetic jets, and a transverse-momentum imbalance. We use the full proton-antiproton collision data set collected by the CDF experiment during the Fermilab Tevatron Run~II at center-of-mass energy $sqrt{s} = 1.96$ TeV, corresponding to an integrated luminosity of 9.1 fb$^{-1}$. A special observable is exploited for an optimal reduction of the dominant systematic uncertainty, associated with the knowledge of the absolute energy of the hadronic jets. The distribution of this observable in the selected events is compared to simulated distributions of ${tbar{t}}$ dilepton signal and background.We measure a value for the top-quark mass of $171.5pm 1.9~{rm (stat)}pm 2.5~{rm (syst)}$ GeV/$c^2$.