This paper studies the mean square stability of a quantized Markov jump linear system (MJLS) that closes its feedback path closed over digital networks. It investigates the effects of the finite constant feedback bit rate on the MJLS' stability. A lower bound is placed on all constant bit rates which could stabilize the MJLS. It also provides a sufficient bit rate for the MJLS, which is shown to be enough to stabilize the system through constructing an appropriate quantization policy. So it is possible to stabilize a quantized MJLS at a finite bit rate. Moreover, it can determine a bit rate range, within which the minimum bit rate required for stability lies. An example is used to verify the achieved theoretical results.