A system forms a cardiac conduction block at a location in a heart of a patient, generally without substantially ablating cardiac tissue. The system includes a delivery system coupled to a source of material that is substantially non-ablative with respect to cardiac tissue but that substantially interrupts and thus blocks cardiac conduction. The delivery system delivers the material to the location, and the material at the location forms a conduction block without substantially ablating the cardiac cells there. The material includes a synthetic polymer, a polysaccharide (e.g. block polysaccharide, alginate, etc.), or a protein, or an analog, derivative, precursor, or agent thereof, or a combination or blend thereof. The material may include living cells. The delivery assembly may include a needle for injecting the material. An expandable member is provided with a needle assembly to deliver the material and form a non-ablative circumferential conduction block where a pulmonary vein extends from an atrium.