Using a sample of dilepton top-quark pair ((tt) over bar) candidate events, a study is performed of the production of top-quark pairs together with heavy-flavor (HF) quarks, the sum of (tt) over bar + b + X and (tt) over bar + c + X, collectively referred to as (tt) over bar + HF. The data set used corresponds to an integrated luminosity of 4.7 fb(-1) of proton-proton collisions at a center-of-mass energy of 7 TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The presence of additional HF (b or c) quarks in the (tt) over bar sample is inferred by looking for events with at least three b-tagged jets, where two are attributed to the b quarks from the (tt) over bar decays and the third to additional HF production. The dominant background to (tt) over bar + HF in this sample is (tt) over bar + jet events in which a light-flavor jet is misidentified as a heavy-flavor jet. To determine the heavy-and light-flavor content of the additional b-tagged jets, a fit to the vertex mass distribution of b-tagged jets in the sample is performed. The result of the fit shows that 79 +/- 14 (stat) +/- 22 (syst) of the 105 selected extra b-tagged jets originate from HF quarks, 3 standard deviations away from the hypothesis of zero (tt) over bar + HF production. The result for extra HF production is quoted as a ratio (R-HF) of the cross section for (tt) over bar + HF production to the cross section for (tt) over bar production with at least one additional jet. Both cross sections are measured in a fiducial kinematic region within the ATLAS acceptance. R-HF is measured to be [6.2 +/- 1.1(stat) +/- 1.8 (syst)]% for jets with p(T) > 25 GeV and vertical bar eta vertical bar < 2.5, in agreement with the expectations from Monte Carlo generators.