One of the open key issues in the astrophysics of stellar coronae is the determination of their spatial structure and density. From almost all previous measurements, one can infer merely the presence of a corona, which for the most energetic stellar coronae may exceed the solar x-ray output by as much as five orders of magnitude, but no information can be obtained on the densities and hence volumes and sizes of the hot x-ray emitting material. A direct spectroscopic measurement of the coronal density was obtained for the star Procyon with the spectrometer on board the Extreme Ultraviolet Explorer satellite; the ratio of two Fe XIV lines at 211.32 and 264.79 angstroms was used to determine a density of similar to 4 x 10(9) to 7 x 10(9) electrons per cubic centimeter, which is a factor of 2 to 3 higher than typical solar active region densities. From this value, we estimate that similar to 6 percent of the stellar surface is covered with similar to 7 x 10(4) coronal loops.