We present the production of dual-species Bose-Einstein condensates (BECs) of 39K and 87Rb. Preparation of both species in the |F=1,mF=−1⟩ state enabled us to exploit a total of three Feshbach resonances, which allows for simultaneous Feshbach tuning of the 39K intraspecies and the 39K−87Rb interspecies scattering length. Thus dual-species Bose-Einstein condensates were produced by sympathetic cooling of 39K with 87Rb. A dark spontaneous force optical trap was used for 87Rb to reduce the losses in 39K due to light-assisted collisions in the optical trapping phase, which can be of benefit for other dual-species experiments. The tunability of the scattering length was used to perform precision spectroscopy of the interspecies Feshbach resonance located at 117.56(2) G and to determine the width of the resonance to 1.21(5) G by rethermalization measurements. The transition region from miscible to immiscible dual-species condensates was investigated and the interspecies background scattering length was determined to 28.5a0 using an empirical model. This paves the way for dual-species experiments with 39K and 87Rb BECs ranging from molecular physics to precision metrology.