The driving distance is 2,609 km - about 1 day 7 hours. See the link below for a map and driving directions.
Further information:
From Cairns, Queensland to Darwin, Northern Territory via the shortest route is a distance of 2369km. It would take an average of 30-31 hours to travel, but this would be without breaks, an impossible feat. Given the conditions under which one would be driving, it is a trip best undertaken over four days.
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