The entropy of a black hole1 and Hawking radiation2 should have the same temperature given by the surface gravity, within a numerical factor of the order of unity. In addition, Hawking radiation should have a thermal spectrum, which creates an information paradox3,4. However, the thermality should be limited by greybody factors5, at the very least6. It has been proposed that the physics of Hawking radiation could be verified in an analogue system7, an idea that has been carefully studied and developed theoretically8,9,10,11,12,13,14,15,16,17,18. Classical white-hole analogues have been investigated experimentally19,20,21, and other analogue systems have been presented22,23. The theoretical works and our long-term study of this subject15,24,25,26,27 enabled us to observe spontaneous Hawking radiation in an analogue black hole28. The observed correlation spectrum showed thermality at the lowest and highest energies, but the overall spectrum was not of the thermal form, and no temperature could be ascribed to it. Theoretical studies of our observation made predictions about the thermality and Hawking temperature29,30,31,32,33. Here we construct an analogue black hole with improvements compared with our previous setup, such as reduced magnetic field noise, enhanced mechanical and thermal stability and redesigned optics. We find that the correlation spectrum of Hawking radiation agrees well with a thermal spectrum, and its temperature is given by the surface gravity, confirming the predictions of Hawking’s theory. The Hawking radiation observed is in the regime of linear dispersion, in analogy with a real black hole, and the radiation inside the black hole is composed of negative-energy partner modes only, as predicted.