This letter considers an unmanned aerial vehicle (UAV)-enabled relay communication system for delivering latency-critical messages with ultra-high reliability, where the relay is operating under amplifier-and-forward (AF) mode. We aim to jointly optimize the UAV location and power to minimize decoding error probability while guaranteeing the latency constraints. Both the free-space channel model and three-dimensional (3-D) channel model are considered. For the first model, we propose a low-complexity iterative algorithm to solve the problem, while globally optimal solution is derived for the case when the signal-to-noise ratio (SNR) is extremely high. For the second model, we also propose a low-complexity iterative algorithm to solve the problem. Simulation results confirm the performance advantages of our proposed algorithms.