It is a very well-known result that in terms of evolutionary stability the long-run outcome of a Cournot oligopoly market with finitely many firms approaches the perfectly competitive Walrasian market outcome (Vega-Redondo, 1997). However, in this paper we show that an asymmetric structure in the cost functions of firms may change the long-run outcome. Contrary to Tanaka (1999) we show that the evolutionarily stable price in an asymmetric Cournot oligopoly needs not equal the marginal cost, it may rather equal a weighted average of (different) marginal cost. We apply a symmetrization technique in order to transform the game with asymmetric firms into a symmetric oligopoly game and then extend Schaff ers definition (1988) of a finite population ESS (FPESS) to this setup. Moreover, we show that the FPESS in this game represents a stochastically stable state of an evolutionary process of imitation with experimentation.