We consider the problem of stabilizing a continuous-time linear time-invariant system subject to communication constraints. A noiseless finite-capacity communication channel connects the process sensors to the controller/actuator. The sensor's state measurements are encoded into symbols from a finite alphabet, transmitted through the channel, and decoded at the controller/actuator. We suppose that the transmission of each symbol costs one unit of communication resources, except for one special symbol in the alphabet that is "free" and effectively signals the absence of transmission. We explore the relationship between the encoder's average bit-rate, its average consumption of communication resources, and the ability of the controller and encoder/decoder pair to stabilize the process. We present a necessary and sufficient condition for the existence of a stabilizing controller and encoder/decoder pair, which depends on the encoder's average bit-rate, its average resource consumption, and the unstable eigenvalues of the process. Moreover, if this condition is satisfied, a stabilizing encoding scheme can be constructed that consumes resources at an arbitrarily small rate, provided the encoder has access to a sufficiently precise clock or large memory. The paper concludes with the analysis of a simple emulation-based controller and event-based encoder/decoder pair that are easy to implement, stabilize the process, and have average bit-rate and resource consumption within a constant factor of the optimal bound.