We describe a large-area detector for measurements of the intensity of cosmic-ray nuclei in balloon-borne exposures. In order to observe individual nuclei at very high energies, the instrument employs transition radiation detectors (TRD) whose energy response extends well beyond 10 4 GeVamu −1 . The TR measurement is performed with arrays of single-wire proportional tubes interleaved with plastic-fiber radiators. An additional energy determination comes from the specific ionization in gas and its relativistic rise which is also measured with proportional tubes. The tubes also determine the trajectory of each cosmic-ray nucleus with mm-resolution. In total, nearly 1600 tubes are used. The instrument is triggered by large-area plastic scintillators. The scintillators, together with acrylic Cherenkov counters, also determine the nuclear charge Z of each cosmic-ray particle, measure the energy in the GeVamu −1 region, and discriminate against low-energy background. We describe the details of this detector system, and discuss its performance in three high-altitude balloon flights, including two long-duration flights in 2003 and 2006 at Antarctic and Arctic latitudes, respectively. Scientific results from these flights are summarized, and possible future developments are reviewed.