A high-resolution grazing-incidence grating spectrometer has been implemented on the Livermore electron beam ion traps for performing very high-resolution measurements in the soft x-ray and extreme ultraviolet region spanning from below 10 Å to above 300 Å. The instrument operates without an entrance slit and focuses the light emitted by highly charged ions located in the roughly 50 μm wide electron beam onto a cryogenically cooled back-illuminated charge-coupled device detector. The measured line widths are below 0.025 Å above 100 Å, and the resolving power appears to be limited by the source size and Doppler broadening of the trapped ions. Comparisons with spectra obtained with existing grating spectrometers show an order of magnitude improvement in spectral resolution.