Using proton-proton collision data corresponding to an integrated luminosity of 3.0  fb^{-1}, recorded by the LHCb detector at center-of-mass energies of 7 and 8 TeV, the B_{c}^{+}→D^{0}K^{+} decay is observed with a statistical significance of 5.1 standard deviations. By normalizing to B^{+}→D[over ¯]^{0}π^{+} decays, a measurement of the branching fraction multiplied by the production rates for B_{c}^{+} relative to B^{+} mesons in the LHCb acceptance is obtained, R_{D^{0}K}=(f_{c}/f_{u})×B(B_{c}^{+}→D^{0}K^{+})=(9.3_{-2.5}^{+2.8}±0.6)×10^{-7}, where the first uncertainty is statistical and the second is systematic. This decay is expected to proceed predominantly through weak annihilation and penguin amplitudes, and is the first B_{c}^{+} decay of this nature to be observed.