An alpha-particle spectrometer has been assembled, consisting of an epitaxial 50 mum thick 4H silicon carbide detector connected to a gallium nitride HEMT used as input transistor of the front-end electronics. The depleted layer of the SiC diode detector was sufficient to stop all alpha particles of the used emitter in the 4.8-MeV to 5.8-MeV energy range. An excellent energy resolution of ~0.9% has been obtained in this energy range at a temperature of 55degC, which is comparable to that of single-crystal diamond detectors. The energy-resolution limiting factor is found to be the dispersion of the energy loss in the gold Schottky contact, which acts as entrance window to the detector. We used a gallium nitride (GaN) front-end transistor because this material offers two important advantages against silicon: (i) it can be grown on SiC substrates so as to realize SiC/GaN integrated systems, (ii) it is a wide band-gap semiconductor and therefore is intrinsically more adequate for room and above-room temperature operation. SiC-detector spectrometers are interesting in many nuclear application where the operation environment is hostile, both in terms of ionising radiation contamination and of high temperatures. Such applications include monitoring of ionising radiations in nuclear power plants and beam diagnostic in fundamental nuclear physics experiments.