This paper studies the joint design over an infinite horizon of the transmission power controller and remote estimator for state estimation over fading channels. A sensor observes a dynamic process and sends its observations to a remote estimator over a wireless fading channel characterized by a time-homogeneous Markov chain. The successful transmission probability depends on both the channel gains and the transmission power used by the sensor. The transmission power control rule and the remote estimator should be jointly designed, aiming to minimize an infinite-horizon cost consisting of the power usage and the remote estimation error. We formulate the joint optimization problem as an average cost belief-state Markov decision process and prove that there exists an optimal deterministic and stationary policy. We then show that when the monitored dynamic process is scalar or the system matrix is orthogonal, the optimal remote estimates depend only on the most recently received sensor observation, and the optimal transmission power is symmetric and monotonically increasing with respect to the norm of the innovation error.