We describe the design of a high-resolution stigmatic extremeultraviolet (EUV) spectroheliometer, configured for flight on a Black Brant sounding rocket, which consists of a 45-cm Gregory telescope coupled to a spectrometer employing a single toroidal diffraction grating in a normalincidence Rowland circle mounting and an imaging pulse-counting multianode microchannel array (MAMA) detector system. The toroidal diffraction grating is fabricated by a technique employing an elastically deformable submaster grating that is replicated in a spherical form and then mechanically distorted to produce the desired aspect ratio of the toroidal surface for stigmatic imaging over the selected wavelength range. The spectroheliometer will produce spatially resolved spectra of the chromosphere, transition region, and corona with an angular resolution of 0.4 arcsec or better, a spectral resolution λ/Δλ of about 104 in first order, and a temporal resolution of the order of seconds. Because of the geometric fidelity of the MAMA detector system, the spectroheliometer will be able to determine Doppler shifts to a resolution of at least 2 mÅ at wavelengths near 600 Å (~1.0 km s-1), depending on the level of the accumulated signal. The unique characteristics of the spectroheliometer will be used in combination with plasma-diagnostic techniques to study the emperature, density, and velocity structures of specific features in the solar outer atmosphere.