A method for measuring weak capture-γ-ray resonances via γγ-coincidence counting techniques is described. The coincidence apparatus consisted of a large-volume germanium detector and an annular NaI(Tl) crystal. The setup was tested by measuring the weak ER=227 keV resonance in 26Mg(p,γ) 27Al. Absolute germanium and NaI(Tl) counting efficiencies for a range of γ-ray energies and for different detector-target geometries are presented. Studies of the γ-ray background in our spectra are described. Compared to previous work, our method improves the detection sensitivity for weak capture-γ-ray resonances by a factor of ≈100. The usefulness of the present technique for investigations of interest to nuclear astrophysics is discussed.