  Cooling microwave resonators to near the quantum ground state, crucial for
their operation in the quantum regime, is typically achieved by direct device
refrigeration to a few tens of millikelvin. However, in quantum experiments
that require high operation power such as microwave-to-optics quantum
transduction, it is desirable to operate at higher temperatures with
non-negligible environmental thermal excitations, where larger cooling power is
available. In this Letter, we present a radiative cooling protocol to prepare a
superconducting microwave mode near its quantum ground state in spite of warm
environment temperatures for the resonator. In this proof-of-concept
experiment, the mode occupancy of a 10-GHz superconducting resonator thermally
anchored at 1.02~K is reduced to $0.44\pm0.05$ by radiatively coupling to a
70-mK cold load. This radiative cooling scheme allows high-operation-power
microwave experiments to work in the quantum regime, and opens possibilities
for routing microwave quantum states to elevated temperatures.
