A kinematically complete experiment devoted to studying the nd-breakup reaction at energies of 8 and 11 MeV via detecting all three secondary particles was performed. The S-1(0) neutron-neutron (nn) scattering-length values of a(nn) = - 19.8 +/- 0.4 and - 19.0 +/- 0.5 fm at, respectively, E-n = 8 and 11 MeV were obtained from a comparison of the experimental dependence of the yield of the nd-breakup reaction on the relative energy of the nn pair with the results of a simulation. An analysis of the values obtained for the nn scattering length, together with data from other experiments, confirms the hypothesis that three-nucleon forces affect the parameters of nn interaction that are extracted from reactions involving few-nucleon systems and gives a new asymptotic nn scattering length, a(nn) = -16.1 +/- 0.1 fm.