Thanks Chester. Just so I'm understanding your solution, you were cutting the nut off perpendicular to the stud? Or were you cutting parallel to the stud to split the nut in half? I have been dousing it with PB Blaster in the meantime, I didn't have a chance to attack it again yet.
After failing perpendicular, I cut parallel to split the nut. I would do the bottom first - if you manage to get the top out, most of the torque you apply is lost to the movement of the sway bar. (And the link just flops around, and hits you in the head).
I first attempted to cut the flange side - thinking the stud and nut would fall out. This didn't work - the flange welded itself on the sway bar and left me with nut side stuck on. (Top tip - pounding on sway bar side is useless because the sway bar absorbs the blow).
So eventually - cutting parallel to the stud allowed me to cut a few relief notches in the nut. (Note that the nuts are hardened, and not easy to split). One I was able to break loose with an air chisel (using the notch I cut as leverage) after jamming a pry bar between the lower control arm and sway bar. It was not easy.
If I had to do it again - I would do what the pro in the thread recommended - break off or melt the link, leaving the ball. (Inside the moon looking thing with blue plastic is a ball that is secured by the nut you see.) Hold the ball with vice grips and turn the nut. (I'm not positive this would have worked in my case, given the amount of rust).
It's just a terrible design. (I did the job twice - failed the first time and took it a shop. The parts the shop installed then failed, so I had to redo it).
Good luck. A face shield and gloves help. Let us know how it works.