We present the first measurement of the angle phi(3) of the unitarity triangle using a model-independent Dalitz plot analysis of B-+/- -> DK +/-, D -> K-S(0)pi(+)pi(-) ecays. The method uses, as input, measurements of the strong phase of the D -> K-S(0)pi(+)pi(-) amplitude from the CLEO Collaboration. The result is based on the full data set of 772 x 10(6) B (B) over bar pairs collected by the Belle experiment at the Gamma(4S) resonance. We obtain phi(3) = (77.3(-14.9)(+15.1) +/- 4.1 +/- 4.3)degrees and the suppressed amplitude ratio r(B) = 0.145 +/- 0.030 +/- 0.010 +/- 0.011. Here the first error is statistical, the second is the experimental systematic uncertainty, and the third is the error due to the precision of the strong-phase parameters obtained by CLEO.