An implantable textile prosthesis includes a first woven section having a tubular branch wall defining a fluid passageway therethrough. The tubular branch wall includes an elongate tubular extent and an asymptotically flared tubular extent, which is asymptotic with respect to the tubular branch wall. The asymptotically flared tubular extent includes a weaving pattern having a plurality of warp yarns and fill yarns and incorporating a gradual change in the number of warp yarns with respect to the fill yarns to provide a seamless and substantially fluid-tight transition region along said flared tubular extent. The tubular branch wall may be sutured to an elongate tubular main wall to provide an implantable tubular textile prosthesis particularly useful in branched end-to-side anastomoses. The tubular main wall may be heat-settably arched to resemble the natural arch of the aorta.