Abstract: Elastic scattering amplitudes for I = 0 DK and I = 0, 1 DK¯ are computed in S, P and D partial waves using lattice QCD with light-quark masses corresponding to mπ = 239 MeV and mπ = 391 MeV. The S-waves contain interesting features including a near-threshold JP = 0+ bound state in I = 0 DK, corresponding to the Ds0∗(2317), with an effect that is clearly visible above threshold, and suggestions of a 0+ virtual bound state in I = 0 DK¯. The S-wave I = 1 DK¯ amplitude is found to be weakly repulsive. The computed finite-volume spectra also contain a deeply-bound D* vector resonance, but negligibly small P -wave DK interactions are observed in the energy region considered; the P and D-wave DK¯ amplitudes are also small. There is some evidence of 1+ and 2+ resonances in I = 0 DK at higher energies.