Apparatus is provided for use with a delivery catheter, including a primary stent-graft and a flared endovascular stent-graft, which is configured to initially be positioned in the delivery catheter in a radially-compressed state, and to assume a radially- expanded state upon being deployed from the delivery catheter. The primary stent-graft includes: a structural member, which includes a plurality of structural stent elements, and which, when the flared endovascular stent-graft assumes the radially-expanded state, is shaped so as to define a flared caudal portion, which flares radially outward in a caudal direction； at least one biologically-compatible substantially fluid-impervious flexible sheet, which is coupled to at least the flared caudal portion； and a stent-engagement member, which is generally tubular when the flared endovascular stent-graft assumes the radially-expanded state, which is disposed at least partially within the flared caudal portion, and which is configured to be sealingly coupled to the primary stent-graft.
