The decay B 0 → DK *0 and the charge conjugate mode are studied using 1.0 fb−1 of pp collision data collected by the LHCb experiment at s√=7s=7 TeV in 2011. The CP asymmetry between the B 0 → DK *0 and the B⎯⎯⎯⎯0→DK⎯⎯⎯⎯∗0B¯0→DK¯∗0 decay rates, with the neutral D meson in the CP-even final state K + K −, is found to be KKd=−0.45±0.23±0.02, AdKK=−0.45±0.23±0.02, where the first uncertainty is statistical and the second is systematic. In addition, favoured B 0 → DK *0 decays are reconstructed with the D meson in the non-CP eigenstate K +π−. The ratio of the B-flavour averaged decay rates in D decays to CP and non-CP eigenstates is measured to be KKd=1.36+0.37−0.32±0.07, RdKK=1.36−0.32+0.37±0.07, where the ratio of the branching fractions of D 0 → K −π+ to D 0 → K + K − decays is included as multiplicative factor. The CP asymmetries measured with two control channels, the favoured B 0 → DK *0 decay with D → K +π− and the B⎯⎯⎯⎯0s→DK∗0B¯s0→DK∗0 decay with D→K + K −, arealsoreported.