We present the results of measurements demonstrating the efficiency of the EDELWEISS-III array of cryogenic germanium detectors for direct dark matter searches. The experimental setup and the FID (Fully Inter-Digitized) detector array is described, as well as the efficiency of the double measurement of heat and ionization signals in background rejection. For the whole set of 24 FID detectors used for coincidence studies, the baseline resolutions for the fiducial ionization energy are mainly below 0.7 keV$_{ee}$ (FHWM) whereas the baseline resolutions for heat energies are mainly below 1.5 keV$_{ee}$ (FWHM). The response to nuclear recoils as well as the very good discrimination capability of the FID design has been measured with an AmBe source. The surface $beta$- and $alpha$-decay rejection power of $R_{rm surf} < 4 times 10^{-5}$ per $alpha$ at 90% C.L. has been determined with a $^{210}$Pb source, the rejection of bulk $gamma$-ray events has been demonstrated using $gamma$-calibrations with $^{133}$Ba sources leading to a value of $R_{gamma{rm -mis-fid}} < 2.5 times 10^{-6}$ at 90% C.L.. The current levels of natural radioactivity measured in the detector array are shown as the rate of single $gamma$ background. The fiducial volume fraction of the FID detectors has been measured to a weighted average value of $(74.6 pm 0.4)%$ using the cosmogenic activation of the $^{65}$Zn and $^{68,71}$Ge isotopes. The stability and uniformity of the detector response is also discussed. The achieved resolutions, thresholds and background levels of the upgraded EDELWEISS-III detectors in their setup are thus well suited to the direct search of WIMP dark matter over a large mass range.
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