Did some googling, and it appears the conductivity of ultra pure water has something to do with the Grotthuss mechanism.
Accordingly, the conductivity of ultra pure water arises from the dissociation of H[sub]2[/sub]O into hydrogen ions (protons or H[sup]+[/sup]) and hydroxide (OH[sup]-[/sup]), which is also an ion.
The proton mobility in water is listed as 3.62 X 10[sup]−3[/sup] cm[sup]2[/sup]/Vs.
So if I understand correctly… if you were somehow able to produce “100% pure water” the Grotthuss mechanism will eventually break up some of the water molecules into protons and hydroxide. The protons are charged (obviously), and mobile, therefore current will result from the application of an electric field and thus the resistance would have a finite value. Not sure about hydroxide… it’s an ion (thus has a charge), but I’m not sure how mobile it is in water. (I tried reading the articles, but quickly discovered they were way over my head.)