We report the fabrication of a 10 nm thick, self-supporting, single-crystal silicon membrane. The fabrication process can be broken up into four major stages. First, a buried SiO2 layer was formed by implantation of oxygen at a depth of 200 nm into a (100) silicon wafer. The size of the membrane was then established by removing the bulk of the silicon over a 1 mm area using a fast acid etch. After this the sample was etched in a hot EDP solution which stops at the buried SiO2 layer. The sample was then cleaned and the SiO2 layers removed, after which it was introduced into a plasma-etching chamber. The membrane was thinned down to a final thickness of 10 nm by RF plasma etching in a gas mixture of carbon tetrafluoride and oxygen. The thickness was monitored during plasma etching by measuring the intensity of He-Ne laser light transmitted through the membrane. The electron energy loss spectrum of the membrane has been measured and shows two features due to single and double plasmon excitation. The plasmon energy was 17.05 eV, in good agreement with previous measurements. Membrane thickness has also been estimated from the area of the plasmon energy loss peak. The final sample had good crystalline quality, was of even thickness over the membrane diameter and showed only a small amount of surface contamination due to the plasma etching stage.