We report a direct measurement of D 0 − ¯¯¯ D 0 mixing parameters through a time-dependent amplitude analysis of the Dalitz plots of D 0 → K 0 S π + π − and, for the first time, D 0 → K 0 S K + K − decays. The low-momentum pion π + s in the decay D * + → D 0 π + s identifies the flavor of the neutral D meson at its production. Using 468.5     fb − 1 of e + e − colliding-beam data recorded near √ s = 10.6     GeV by the BABAR detector at the PEP-II asymmetric-energy collider at SLAC, we measure the mixing parameters x = [ 1.6 ± 2.3 ( stat ) ± 1.2 ( syst ) ± 0.8 ( model ) ] × 10 − 3 , and y = [ 5.7 ± 2.0 ( stat ) ± 1.3 ( syst ) ± 0.7 ( model ) ] × 10 − 3 . These results provide the best measurement to date of x and y . The knowledge of the value of x , in particular, is crucial for understanding the origin of mixing.