We demonstrate radiation-pressure cavity-cooling of a mechanical mode of a micromirror starting from cryogenic temperatures. To achieve that, a high-finesse Fabry-Pérot cavity (F≈2200) was actively stabilized inside a continuous-flow 4He cryostat. We observed optical cooling of the fundamental mode of a 50 μm×50 μm×5.4 μm singly clamped micromirror at ωm=3.5 MHz from 35 K to approximately 290 mK. This corresponds to a thermal occupation factor of ⟨n⟩≈1×104. The cooling performance is only limited by the mechanical quality and by the optical finesse of the system. Heating effects, e.g. due to absorption of photons in the micromirror, could not be observed. These results represent a next step towards cavity-cooling a mechanical oscillator into its quantum ground state.