A method of creating an AV fistula between adjacent first and second blood vessels, includes steps of cutting a hole through the adjacent walls of the first and second blood vessels and inserting a welding catheter into the first vessel, and through the hole into the second vessel, so that a distal end of the welding catheter is disposed within the second vessel. A portion of wall defining the welding catheter is then expanded radially outwardly, and the expanded wall portion is pulled proximally to engage the wall of the second blood vessel and to pull it toward the wall of the first blood vessel. Then, a portion of wall proximal to the first expanded wall portion and disposed in the first blood vessel is expanded radially outwardly, thereby capturing the walls of each of the second and first blood vessels between the two expanded wall portions. Cutting elements on the expanded wall portions are then energized to create a tissue welded elongate aperture between the first and second blood vessels.
