Most of the plutonium in the world resides inside spent nuclear reactor fuel rods. This high-level radioactive waste is commonly held in long-term storage within large, heavily shielded casks. Currently, international nuclear safeguards inspectors have no stand-alone method of verifying the amount of reactor fuel stored within a sealed cask. Here we demonstrate experimentally that measurements of the scattering angles of cosmic ray muons which pass through a storage cask can be used to determine if spent fuel assemblies are missing without opening the cask. This application of technology and methods commonly used in high-energy particle physics provides a potential solution to this long-standing problem in international nuclear safeguards.