The highly-stable Q-switched longitudinally diode-pumped microchip laser, emitting radiation at wavelength 1031 nm, was designed and realized. This laser was based on monolith crystal which combines in one piece an active laser part (YAG crystal doped with Yb3+ ions, 10 at.% Yb/Y, 3mm long) and saturable absorber (YAG crystal doped with Cr3+ ions, 1.36mm long). The diameter of the diffusion bonded monolith was 3 mm. The initial transmission of the Cr:YAG part was 90% @ 1031 nm. The microchip resonator consisted of dielectric mirrors directly deposited on the monolith surfaces. The pump mirror (HT for pump radiation, HR for generated radiation) was placed on the Yb:YAG part. The output coupler with reflection 55% for the generated wavelength was placed on the Cr3+-doped part. Q-switched microchip laser was tested under CW diode pumping. For longitudinal pumping of Yb:YAG part, a fibre coupled (core diameter 100 μm, NA= 0.22) laser diode, operating at wavelength 968 nm, was used. The laser threshold was 3.3W. The laser slope efficiency calculated for output mean power in respect to incident CW pumping was 17%. The wavelength of linearly polarized laser emission was fixed to 1031 nm. The generated transversal intensity beam profile was close to the fundamental Gaussian mode. The generated pulse length was equal to 1.6 ns (FWHM). This value was mostly stable and independent on investigated pumping powers in the range from the threshold up to 9.3W. The single pulse energy was linearly increasing with the pumping power. Close to the laser threshold the generated pulse energy was 45 μJ. For maximum investigated CW pumping 9.3W the pulse energy was stabilized to 74 μJ which corresponds to the Q-switched pulse peak power 46 kW. The corresponding Q-switched pulses repetition rate was 13.6 kHz. The maximum Yb:YAG/Cr:YAG microchip laser mean output power of 1W was reached without observable thermal roll-over.