Close-packed nanoparticle monolayers have recently been shown to form mechanically robust, free-standing membranes. We report the first measurements of molecular transport through such ultrathin sheets, self-assembled from dodecanethiol-ligated gold nanocrystals. For aqueous solutions we find filtration coefficients 2 orders of magnitude larger than those observed in polymer-based filters, sieving of large solutes, and for smaller solutes a pronounced dependence of rejection on being charged. These results open up new possibilities for controlled delivery and separation of nano-objects.