Arterial vein grafts (AVGs) often fail due to intimal hyperplasia, thrombosis, or accelerated atherosclerosis. Various approaches have been proposed to address AVG failure, including delivery of temporary mechanical support, many of which could be facilitated by perivascular placement of a biodegradable polymer wrap. The purpose of this work was to demonstrate that a polymer wrap can be applied to vein segments without compromising viability/function, and to demonstrate one potential application, i.e., gradually imposing the mid-wall circumferential wall stress (CWS) in wrapped veins exposed to arterial levels of pressure. Poly(ester urethane)urea, collagen, and elastin were combined in solution, and then electrospun onto freshly-excised porcine internal jugular vein segments. Tissue viability was assessed via Live/Dead staining for necrosis, and vasomotor challenge with epinephrine and sodium nitroprusside for functionality. Wrapped vein segments were also perfused for 24h within an ex vivo vascular perfusion system under arterial conditions (pressure = 120/80 mmHg; flow = 100 mL/min), and CWS was calculated every hour. Our results showed that the electrospinning process had no deleterious effects on tissue viability, and that the mid-wall CWS vs. time profile could be dictated through the composition and degradation of the electrospun wrap. This may have important clinical applications by enabling the engineering of an improved AVG.