Correlation does not imply causation, however correlation can be used to support a theory of causation. The article does address this point:

   "There is a substantial causal relationship," she says.
   "I can see it in the state-to-state variations. States
   that experienced particularly early or particularly sharp
   declines in lead experienced particularly early or 
   particularly sharp declines in violent crime 20 years later."

Without reading the paper, I will not venture any opinion on the hypothesis/research/etc itself. I do however thing it may take more than 5 words to completely dismiss the theory.