  We measure the branching fractions and CP asymmetries in the decays B0 -->
K+K-K0S and B+ --> K+K0S K0S using a sample of approximately 122 million BBbar
pairs collected by the BaBar detector. From a time-dependent analysis of the K+
K- K0S sample that excludes phi K0S, the values of the CP-violation parameters
are S = -0.56 +/- 0.25 +/- 0.04 and C = -0.10 +/- 0.19 +/- 0.10, where the
first uncertainty is statistical, the second is systematic. We confirm that the
final state is nearly purely CP-even. Using this result and setting C=0, we
extract the Standard Model parameter sin2beta = 0.57 +/- 0.26 +/- 0.04 +0.17/-0
where the last error is due to uncertainty on the CP content. We present the
first measurement of the CP-violating charge asymmetry ACP(B+ --> K+K0S K0S) =
-0.04 +/- 0.11 +/- 0.02, with a 90% confidence-level interval of [-0.23, 0.15].
The branching fractions are BF(B0 --> K+K-K0) = (23.8 +/- 2.0 +/- 1.6) x
10^{-6} and BF(B+ --> K+K0S K0S) = (10.7 +/- 1.2 +/- 1.0) x 10^{-6}.
