Oh duh, I figured out the problem. I was keying something else (not the weight) What exactly does "WO" stand for in the channel box under a constraint? And what does the 1 value after it signify?
The odd thing is that, when I create a constraint and set the weight to zero in the options box, the constraint is still created with a starting weight of 1?
Don't know why that is...
Anyway, thanks Xander for help. Turned out it was just my ignorance! (as usual)