Elementary! The butler, knowing that he was rumbled, stole the Tiki idol from the weighted pedestal, and positioned the duke in front of the blowdarts. I was able to deduce this from the clay on his shoes, that is only found in hidden temples in the deepest darkest regions of the world, and handed him over to Inspector Lestrade. Pip pip.

However I think the real mystery now is why Mulder didn't smite Scully for not believing in any of the stuff he did.