The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot. The stable concordance genus describes the behavior of the concordance genus under connected sum, and can be a valuable tool in calculating the concordance genus for certain families of knots. We will present several computations of the stable concordance genus and give a realization result.