The thermal conductance of a single channel is limited by its unique quantum value G_Q, as was shown theoretically¹ in 1983. This result closely resembles the well-known quantization of electrical conductance in ballistic one-dimensional conductors^{2, 3}. Interestingly, all particles—irrespective of whether they are bosons or fermions—have the same quantized thermal conductance^{4, 5} when they are confined within dimensions that are small compared to their characteristic wavelength. The single-mode heat conductance is particularly relevant in nanostructures. Quantized heat transport through submicrometre dielectric wires by phonons has been observed⁶, and it has been predicted to influence cooling of electrons in metals at very low temperatures due to electromagnetic radiation⁷. Here we report experimental results showing that at low temperatures heat is transferred by photon radiation, when electron–phonon⁸ as well as normal electronic heat conduction is frozen out. We study heat exchange between two small pieces of normal metal, connected to each other only via superconducting leads, which are ideal insulators against conventional thermal conduction. Each superconducting lead is interrupted by a switch of electromagnetic (photon) radiation in the form of a DC-SQUID (a superconducting loop with two Josephson tunnel junctions). We find that the thermal conductance between the two metal islands mediated by photons indeed approaches the expected quantum limit of G_Q at low temperatures. Our observation has practical implications—for example, for the performance and design of ultra-sensitive bolometers (detectors of far-infrared light) and electronic micro-refrigerators⁹, whose operation is largely dependent on weak thermal coupling between the device and its environment