We present a new measurement of the unitarity triangle angle phi(3) using a Dalitz plot analysis of the K-S(0)pi(+)pi(-) decay of the neutral D meson produced in B-+/- -> D-(()) K-+/- decays. The method exploits the interference between D-0 and (D) over bar (0) to extract the angle phi(3), strong phase delta, and the ratio r of suppressed and allowed amplitudes. We apply this method to a 605 fb(-1) data sample collected by the Belle experiment. The analysis uses three decays: B-+/- -> DK +/-, and B-+/- -> DK-+/- with D* -> D pi(0) and D* -> D gamma, as well as the corresponding charge-conjugate modes. From a combined maximum likelihood fit to the three modes, we obtain phi(3) = 78.4 degrees (+10.8 degrees)(-11.6 degrees) +/- 3.6 degrees (syst) +/- 8.9 degrees (model). CP conservation in this process is ruled out at the confidence level (1 - CL) = 5 X 10(-4), or 3.5 standard deviations.