Abstract Accuracy of biological discrimination at the molecular level is known in some systems to involve kinetic proofreading mechanisms. Hopfield and Ninio were the first to propose simple specific kinetic mechanisms for such proofreading and to demonstrate that an energy cost accompanies their improvement in accuracy. Savageau and Freter subsequently derived the explicit cost-accuracy relationship for a broad class of proofreading mechanisms, including the conventional Hopfield-Ninio mechanism just referred to. In other systems, the presence of proofreading mechanisms is in question because the diagnostic features of conventional kinetic proofreading are absent. However, Hopfield has recently proposed an alternative “energy-relay” mechanism, which lacks the characteristic features of conventional proofreading and yet is capable of improving accuracy. In this paper, I use the general cost-accuracy relationship that we have previously derived to examine the energy cost and accuracy of proofreading mechanisms involving an energy relay. The principal findings are the following. First, such mechanisms improve accuracy with a zero cost of proofreading, when “proofreading cost,” defined as the cost due specifically to proofreading, is separated from the costs of putting material through the system. Second, the basic energy-relay mechanism discussed by Hopfield has only a modest improvement in accuracy, but a comparable improvement by a conventional proofreading mechanism would have a cost of about 0·0352 (moles ATP per mole of total product output). Third, accuracy can be increased somewhat if multiple stages of conventional kinetic proofreading precede the energy-relay mechanism. The cost for this improvement is zero while a comparable increase in accuracy achieved by conventional proofreading alone has a cost of about 0·0385. Finally, I propose an alternative arrangement of energy-relay mechanisms that is capable of increasing accuracy still further. The maximum accuracy achieved by this scheme at zero energy cost is comparable to that achieved by an infinite expenditure of energy in a single stage of conventional proofreading.