A novel concentric spectrometer having one convex grating and one concave mirror, working at ${+}{1}$+1st order diffraction, and with a small size, high resolution, and high diffraction efficiency, is proposed. It can simultaneously achieve high resolution and compactness by increasing the grating groove density. A compact spectrometer operating at a wavelength of 740-790 nm with an excellent imaging quality is designed. Its spectral resolution reaches 0.049 nm, and its diffraction efficiency improves by 27% compared to the conventional Offner spectrometer with convex grating working at ${-}{1}$-1st order diffraction. This is suitable for small, light, and low-cost atmospheric gas monitoring satellites.