We propose a method of generating a vortex ring in a Bose-Einstein condensate by means of electromagnetically-induced atomic transitions. Some remnant population of atoms in a second internal state remains within the toroidal trap formed by the mean field repulsion of the vortex ring. This population can be removed, or it can be made to flow around the torus (i.e. within the vortex ring). If this flow has unit topological winding number, the entire structure formed by the two condensates is an example of a three-dimensional skyrmion texture.