A problem concerning the stochastic stability of linear systems with a random time-delay is discussed. The delay is assumed to be a stationary and ergodic process. It is shown that the system is almost surely a symptotically stable, if 1) there exists a positive constant such that a system in which the delay is fixed to be the constant is exponentially stable and 2) the first absolute moment of the delay around the constant is sufficiently small. An example is presented to explain the practical application of the result.