An extreme ultraviolet spectrometer has been developed for high-energy density physics experiments at the Shenguang-III (SG-III) laser facility. Alternative use of two different varied-line-spacing gratings covers a wavelength range of 10-260 Å. A newly developed x-ray framing camera with single wide strip line is designed to record time-gated spectra with ~70 ps temporal resolution and 20 lp/mm spatial resolution. The width of the strip line is up to 20 mm, enhancing the capability of the spatial resolving measurements. All components of the x-ray framing camera are roomed in an aluminum air box. The whole spectrometer is mounted on a diagnostic instrument manipulator at the SG-III laser facility for the first time. A new alignment method for the spectrometer based on the superimposition of two laser focal spots is developed. The approaches of the alignment including offline and online two steps are described. A carbon spectrum and an aluminum spectrum have been successfully recorded by the spectrometer using 2400 l/mm and 1200 l/mm gratings, respectively. The experimental spectral lines show that the spectral resolution of the spectrometer is about 0.2 Å and 1 Å for the 2400 l/mm and 1200 l/mm gratings, respectively. A theoretical calculation was carried out to estimate the maximum resolving power of the spectrometer.