This paper presents test results on a prototype superconducting undulator magnet fabricated using 15% Zr-doped rare-earth barium copper oxide high temperature superconducting (HTS) tapes. On an 11-pole magnet we demonstrate an engineering current density, Je, of more than 2.1 kA mm−2 at 4.2 K, a value that is 40% higher than reached in comparable devices wound with NbTi-wire, which is used in all currently operating superconducting undulators. A novel winding scheme enabling the continuous winding of tape-shaped conductors into the intricate undulator magnets as well as a partial interlayer insulation procedure were essential in reaching this advance in performance. Currently, there are rapid advances in the performance of HTS; therefore, achieving even higher current densities in an undulator structure or/and operating it at temperatures higher than 4.2 K will be possible, which would substantially simplify the cryogenic design and reduce overall costs.