We report on the generation of continuous-wave, intra-cavity frequency-doubled, multi-mode laser radiation in an Yb:LuAG thin-disk laser. Output powers of up to 1 kW at a wavelength of 515 nm were achieved at an unprecedented optical efficiency of 51.6\% with respect to the pumping power of the thin-disk laser. The wavelength stabilization and spectral narrowing as well as the polarization selection, which is necessary for a stable and efficient second-harmonic generation, was achieved by the integration of a diffraction grating into the dielectric end mirror of the cavity, which exhibits a diffraction efficiency of 99.8\%. At a frequency-doubled output power of 820 W the peak-to-valley power fluctuations measured during 100 minutes of laser operation amounted to only 8.2 W (1.0\%). The beam parameter product of the frequency-doubled output was 3.4 mm·mrad (M2 $\approx$ 20), which is suitable for standard beam delivery using fibers with a core diameter of 100 µm and a NA of 0.2.