Abstract: A method and apparatus for providing permanent arterial-venous (“AV”) access to a patient for the purposes of conducting hemodialysis procedures and the like. In one embodiment, an AV graft is provided with a venous end of relatively small diameter adapted to be introduced into a patient's superior vena cava. The opposite, arterial end of the graft has a larger diameter than the venous end, while an intermediate body portion of the graft has an even larger diameter. In one configuration, the graft is introduced using a cut-down procedure in which venous access is established at a location on the patient's arm, while the venous end of the graft is fed through the patient's vein until the distal venous end thereof becomes situated in the superior vena cava. The arterial end of the graft is then tunneled down the patient's arm to a desired arterial access site, where an anastomotic coupling is effectuated.