We report laboratory measurements of the absorption coefficient of solid para-H2, within the wavelength range from 1 to 16.7 micron, at high spectral resolution. In addition to the narrow rovibrational lines of H2 which are familiar from gas phase spectroscopy, the data manifest double transitions and broad phonon branches that are characteristic specifically of hydrogen in the solid phase. These transitions are of interest because they provide a spectral signature which is independent of the impurity content of the matrix. We have used our data, in combination with a model of the ultraviolet absorptions of the H2 molecule, to construct the dielectric function of solid para-H2 over a broad range of frequencies. Our results will be useful in determining the electromagnetic response of small particles of solid hydrogen. The dielectric function makes it clear that pure H2 dust would contribute to IR extinction predominantly by scattering starlight, rather than absorbing it, and the characteristic IR absorption spectrum of the hydrogen matrix itself will be difficult to observe.