We consider a basic joint communication and sensing setup comprising a transmitter, a receiver and a sensor. The transmitter sends a codeword to the receiver through a discrete memoryless channel, and the receiver is interested in decoding the transmitted codeword. At the same time, the sensor picks up a noisy version of the transmitted codeword through one of two possible discrete memoryless channels. The sensor knows the codeword and wishes to discriminate between the two possible channels, i.e. to identify the channel that has generated the output given the input. We study the trade-off between communication and sensing in the asymptotic regime, captured in terms of the channel coding rate against the two types of discrimination error exponents. We characterize the optimal trade-off between the rate and the exponents for general discrete memoryless channels with an input cost constraint.