What is the maximum rate at which digital information can be communicated without error using electromagnetic signals, such as radio communication? According to Shannon theory this rate is the capacity of the communication channel, which is obtained by maximizing the mutual information between the channel’s input and output. Shannon theory, however, has been developed within classical physics, whereas electromagnetic signals are, ultimately, quantum-mechanical entities. To account for this fact, the capacity must be expressed in terms of a complicated optimization of the Holevo information, but explicit solutions are still unknown for arguably the most elementary electromagnetic channel, the one degraded by additive thermal noise. We place bounds on the thermal channel’s Holevo information that determine the capacity up to corrections that are insignificant for practical scenarios such as those with high noise or low transmissivity. Our results apply to any bosonic thermal-noise channel, including electromagnetic signalling at any frequency.