The low-background, VUV-sensitive 3-inch diameter photomultiplier tube R11410 has been developed by Hamamatsu for dark matter direct detection experiments using liquid xenon as the target material. We present the results from the joint effort between the XENON collaboration and the Hamamatsu company to produce a highly radio-pure photosensor (version R11410-21) for the XENON1T dark matter experiment. After introducing the photosensor and its components, we show the methods and results of the radioactive contamination measurements of the individual materials employed in the photomultiplier production. We then discuss the adopted strategies to reduce the radioactivity of the various PMT versions. Finally, we detail the results from screening 286 tubes with ultra-low background germanium detectors, as well as their implications for the expected electronic and nuclear recoil background of the XENON1T experiment.