A system and method of use for effecting the bypass or other anastomosis of a portion of a native blood vessel, duct, lumen or other tubular organ within the body of a living being. The system includes an anastomosis connector device and a deployment instrument for carrying the device to the desired position within the vessel, duct, lumen or tubular organ. The deployment device may include a dilator to facilitate the placement of the anastomosis device within the interior of the vessel, duct, lumen or other tubular organ. The anastomosis device is preferably formed of a resorbable material and is configured to minimize blood or other fluid turbulence therethrough. The device may include snap-connectors or other components for securing it to the tissue of the vessel, duct, lumen or tubular organ and hemostasis-inducing sealing rings to prevent blood leakage. Other components may be included in the device for expediting the anastomosis procedure, with or without the use of sutures.