This work concerns zero-sum stochastic games with finite state space and compact action sets. The game is driven by two players and it is assumed that player 1 has a nonull and constant risk-sensitivity coefficient, so that a random cost is assessed via an exponential disutility function. The main objective of the paper is to show that as the discount factor increases to 1, an appropriate normalization of the risk-sensitive discounted value function converges to the risk-sensitive average value function. Besides standard continuity-compactness conditions, that result is obtained assuming that (i) the state space is communicating under the action of any pair of stationary deterministic policies and (ii) a minorization property holds at some state. Additionally, it is shown that, if as in the classical risk-neutral context, the risk-sensitive discounted value function is multiplied by 1 minus the discount factor, then the resulting normalization converges to the arithmetic mean of the risk-sensitive average value over the interval joining 0 and the risk-sensitivity coefficient.