  The selection of low-radioactive construction materials is of utmost
importance for the success of low-energy rare event search experiments. Besides
radioactive contaminants in the bulk, the emanation of radioactive radon atoms
from material surfaces attains increasing relevance in the effort to further
reduce the background of such experiments. In this work, we present the
$^{222}$Rn emanation measurements performed for the XENON1T dark matter
experiment. Together with the bulk impurity screening campaign, the results
enabled us to select the radio-purest construction materials, targeting a
$^{222}$Rn activity concentration of 10 $\mu$Bq/kg in 3.2 t of xenon. The
knowledge of the distribution of the $^{222}$Rn sources allowed us to
selectively eliminate critical components in the course of the experiment. The
predictions from the emanation measurements were compared to data of the
$^{222}$Rn activity concentration in XENON1T. The final $^{222}$Rn activity
concentration of (4.5 $\pm$ 0.1) $\mu$Bq/kg in the target of XENON1T is the
lowest ever achieved in a xenon dark matter experiment.
