We consider a communication model in which a transmitter attempts to communicate with a receiver over a state-dependent channel and simultaneously estimate the state using strictly causal noisy state observations. Motivated by joint communication and sensing scenarios in which the physical phenomenon of interest for sensing evolves at a much slower rate than the rate of communication, the state is assumed to remain constant over the duration of the transmission. We derive a complete characterization of the optimal asymptotic trade-off between communication rate and detection-error exponent when coding strategies are open loop. We also show that closed-loop strategies result in strict improvements of the trade-offs.