The existence of the cosmic microwave background radiation is a fundamental prediction of hot Big Bang cosmology, and its temperature should increase with increasing redshift. At the present time (redshift z = 0), the temperature has been determined with high precision to be TCMBR(0) = 2.726 +/- 0.010K. In principle, the background temperature can be determined using measurements of the relative populations of atomic fine-structure levels, which are excited by the background radiation. But all previous measurements have achieved only upper limits, thus still formally permitting the radiation temperature to be constant with increasing redshift. Here we report the detection of absorption lines from the first and second fine-structure levels of neutral carbon atoms in an isolated cloud of gas at z = 2.3371. We also detected absorption due to several rotational transitions of molecular hydrogen, and fine-structure lines of singly ionized carbon. These constraints enable us to determine that the background radiation was indeed warmer in the past: we find that TCMBR(z = 2.3371) is between 6.0 and 14K. This is in accord with the temperature of 9.1K predicted by hot Big Bang cosmology.