A tissue ablation device creates long linear lesions along a body space wall of an animal, and primarily between adjacent pulmonary vein ostia in a left atrial wall. An ablation element includes first and second ends that are bordered by first and second anchors. The anchors are adapted to secure the ablation element ends at predetermined first and second locations along the body space wall such that the ablation element is adapted to ablate an elongate region of tissue between those locations. The anchors may be guidewire tracking members, each including a bore adapted to receive and track over a guidewire, and may anchor within adjacent pulmonary vein ostia when the engaged guidewires are positioned within the respective veins. Stop members may be provided on the guidewires and may be adapted for positioning the relative anchors or for forcing the anchors to fit snugly within the vein ostia. A conduit passageway through the catheter houses a stiffening stylet which may be advanced into the region of the ablation element in order to impart a shape to that element to conform it to a predetermined region of anatomy, or to stiffen the underlying catheter in order to advance the assembly into remote anatomy.