We present a study of the decays ¯¯¯ B 0 → D ( * ) 0 ¯¯¯¯ K ( * ) 0 using a sample of 226 × 10 6 Υ ( 4 S ) → B ¯¯¯ B decays collected with the BABAR detector at the PEP-II asymmetric-energy e + e − collider at SLAC. We report evidence for the decay of B 0 and ¯¯¯ B 0 mesons to the D * 0 K 0 S final state with an average branching fraction B ( B 0 → D * 0 K 0 ) ≡ ( B ( ¯¯¯ B 0 → D * 0 ¯¯¯¯ K 0 ) + B ( B 0 → D * 0 K 0 ) ) / 2 = ( 3.6 ± 1.2 ± 0.3 ) × 10 − 5 . Similarly, we measure B ( B 0 → D 0 K 0 ) ≡ ( B ( ¯¯¯ B 0 → D 0 ¯¯¯¯ K 0 ) + B ( B 0 → D 0 K 0 ) ) / 2 = ( 5.3 ± 0.7 ± 0.3 ) × 10 − 5 for the D 0 K 0 S final state. We measure B ( ¯¯¯ B 0 → D 0 ¯¯¯¯ K * 0 ) = ( 4.0 ± 0.7 ± 0.3 ) × 10 − 5 and set a 90% confidence level upper limit B ( ¯¯¯ B 0 → ¯¯¯ D 0 ¯¯¯¯ K * 0 ) < 1.1 × 10 − 5 . We determine the upper limit for the decay amplitude ratio | A ( ¯¯¯ B 0 → ¯¯¯ D 0 ¯¯¯¯ K * 0 ) / A ( ¯¯¯ B 0 → D 0 ¯¯¯¯ K * 0 ) | to be less than 0.4 at the 90% confidence level.