A membrane of multiwall carbon nanotubes embedded in a silicon nitride matrix was fabricated for use in studying fluid mechanics on the nanometer scale. Characterization by fluorescent tracer diffusion and scanning electron microscopy suggests that the membrane is void-free near the silicon substrate on which it rests, implying that the hollow core of the nanotube is the only conduction path for molecular transport. Assuming Knudsen diffusion through this nanotube membrane, a maximum helium transport rate (for a pressure drop of 1 atm) of 0.25 cc/sec is predicted. Helium flow measurements of a nanoporous silicon nitride membrane, fabricated by sacrificial removal of carbon, give a flow rate greater than 1×10-6 cc/sec. For viscous, laminar flow conditions, water is estimated to flow across the nanotube membrane (under a 1 atm pressure drop) at up to 2.8×10-5 cc/sec (1.7 μL/min).