The Normal Incidence X-Ray Telescope (NIXT) obtained a unique set of high-resolution full-disk solar images which were exposed simultaneousy by X-rays in a passband at 63.5 Angstrom and by visible light. The perfect alignment of a photospheric visible-light image with a coronal X-ray image enables us to present observations of X-ray intensity as a function of an accurately determined height above the visible limb. The height at which the observed X-ray intensity peaks varies from 4000 km in active regions to 9000 km in quiet regions of the Sun. The interpretation of the observations stems from the previously established fact that, for the coronal loops, emission in the NIXT bandpass peaks sharply just above the footpoints. Because there is not a sharp peak in the observed X-ray intensity as a function of off-limb height, we conclude that the loop footpoints, when viewed at the limb, are obscured by absorption in chromospheric material along the line of sight. We calculate the X-ray intensity as a function of height predicted by a number of different idealizations of the solar atmosphere, and we compare these calculations with the observed X-ray intensity as a function of height. The calculations use existing coronal and chromospheric models. In order for the calculations to reproduce the observed off-limb X-ray intensities, we are forced to assume an atmosphere in which the footpoints of coronal loops are interspersed along the line of sight with cooler chromospheric material extending to heights well above the loop footpoints. We argue that the absorption coefficient for NIXT X-rays by chromospheric material is roughly proportional to the neutral hydrogen density, and we estimate an average neutral hydrogen density and scale height implied by the data.