Depending on the strength of the spell (Cure .vs. Cura .vs. Curaga, in theory): It's pretty much entirely effective if they drink only sealed bottled water or juice/soda/etc from that point onward. The illness doesn't reoccur. A plain old Cure probably won't get rid of all of it, but anything stronger (Cura, Curaga) will do the trick. :)

(If we're not referring to Final Fantasy's Cure spell here, just pretend I said "weak, moderate, and strong Cure spells" instead.)