The feasibility of welding a CuAlBe SMA by continuous drive friction welding was evaluated. The metallurgical state (annealed/quenched) before welding and frictional pressure (5 and 10 MPa) were varied and their effects on joint quality were analyzed. Static tensile tests, microhardness, thermal analysis by DSC, and optical microscopy were carried out to characterize the welded joint. The results indicated joints of excellent thermomechanical quality. The welding zones are well-defined, narrow, and have a very refined microstructure compared to the base metal. The phase transition temperatures along the welded assemblies were not changed when welding was performed on the quenched samples, except in the welding zone of the sample welded with 10 MPa. Maximum tensile strength was obtained by using maximum friction pressure during welding of the annealed alloy (quenching after welding). To fill the gap in bibliographical research in this field of study, this work innovatively presents the possibility of welding Cu-based SMAs by rotary friction, including the welding of quenched parts without the need for subsequent heat treatments and without compromising the shape memory effect.