The diffractive dissociation of a 200-GeV/c π− beam into KS0KS0π+π−π− has been observed. The diffractive KS0KS0π+π−π− cross section is 1.59±0.78 μb. The ratio of the diffractive KS0KS0π+π−π− cross section to the diffractive KS0KS0π− cross section is 0.40±0.13, which is in good agreement with a diffractive-fragmentation-model prediction of 0.36. There is evidence for simultaneous production of K*− and K*+ in the diffractive KS0KS0π+π−π− sample. The K*+−KS0π−+ mass distribution shows an enhancement near 1.95 GeV.