This paper addresses the normative issue of how tax rates should be set to control a Cournot duopoly under asymmetric information. If the marginal social value of one firm's action depends on the action of the other, taxes should be based upon the actions of both. The qualitative efficiency results of monopoly regulation carry over in a conditional Cournot sense. When pre-tax profits are independent, it is possible to implement the optimal action choices in a decentralised regulatory system, though aggregate welfare is lower. This result is used to solve the general duopoly problem when the information parameter is continuous.