When does "X is NP-complete" imply "#X is #P-complete"?
The most recent paper on this question seems to be:

Noam Livne, A note on #P-completeness of NP-witnessing relations, 
Information Processing Letters, Volume 109, Issue 5, 15 February 2009, Pages 259–261
http://www.sciencedirect.com/science/article/pii/S0020019008003141

which gives some sufficient conditions.

Interestingly the introduction states "To date, all known NP complete sets have a defining relation which is #P complete", so the answer to Suresh's comment is "no examples are known".