The cross sections of nuclear reactions between the radioisotope ^{7}Be and deuterium, a possible mechanism of reducing the production of mass-7 nuclides in big-bang nucleosynthesis, were measured at center-of-mass energies between 0.2 and 1.5 MeV. The measured cross sections are dominated by the (d,α) reaction channel, towards which prior experiments were mostly insensitive. A new resonance at 0.36(5) MeV with a strength of ωγ=1.7(5) keV was observed inside the relevant Gamow window. Calculations of nucleosynthesis outcomes based on the experimental cross section show that the resonance reduces the predicted abundance of primordial ^{7}Li, but not sufficiently to solve the primordial lithium problem.