The turbulent thermal convection of a horizontal plane layer of electrically conducting Boussinesq fluid is considered. It is permeated by a horizontal magnetic field, which is maintained by dynamo action unaided by any externally applied electric currents. Upper bounds for the ohmic dissipation in the mean magnetic fieldMare obtained by maximisingMitself subject to a number of constraints which include power integrals derived from the governing equations. The structure of the optimal vector field depends on the magnitude of both the Rayleigh number R and a magnetic Prandtl numberqwhich is assumed to be small. Nevertheless irrespective of the value ofq, it is found thatMis bounded above according to the universal lawMℳKR3/2, whereKis a constant whose numerical value is independent of the parameters characterising the problem.