A stent cover of elastic material having a very precisely defined pattern of elongated perforations formed therein so as to control the expansion of the stent to a preselected shape. Once in place and expanded, the cover additionally prevents the prolapse of lumen wall tissue as well as the escape of embolic materials. The pattern of perforations opens into a pattern of openings that promote the proliferation of endothelial tissue and hence the healing of the lumen at the deployment site.