  The XENON1T experiment aims for the direct detection of dark matter in a
cryostat filled with 3.3 tons of liquid xenon. In order to achieve the desired
sensitivity, the background induced by radioactive decays inside the detector
has to be sufficiently low. One major contributor is the $\beta$-emitter
$^{85}$Kr which is an intrinsic contamination of the xenon. For the XENON1T
experiment a concentration of natural krypton in xenon $\rm{^{nat}}$Kr/Xe < 200
ppq (parts per quadrillion, 1 ppq = 10$^{-15}$ mol/mol) is required. In this
work, the design of a novel cryogenic distillation column using the common
McCabe-Thiele approach is described. The system demonstrated a krypton
reduction factor of 6.4$\cdot$10$^5$ with thermodynamic stability at process
speeds above 3 kg/h. The resulting concentration of $\rm{^{nat}}$Kr/Xe < 26 ppq
is the lowest ever achieved, almost one order of magnitude below the
requirements for XENON1T and even sufficient for future dark matter experiments
using liquid xenon, such as XENONnT and DARWIN.
