Information propagation by sequence-specific, template-catalyzed molecular assembly is the source of the biochemical complexity of living systems, allowing thousands of sequence-defined proteins to be produced from only 20 distinct building blocks. By contrast, exploitation of this powerful chemical motif is rare in non-biological contexts, particularly enzyme-free environments, with even the template-catalyzed formation of dimers being a significant challenge. The main obstacle is product inhibition: the tendency of products to bind to their templates more strongly than individual monomers, preventing effective catalytic templating of longer polymers. Here we present a rationally designed enzyme-free system in which a DNA template catalyzes, with weak competitive product inhibition, the production of sequence-specific DNA dimers. We demonstrate the selective templating of 9 different dimers with high specificity and catalytic turnover. Most importantly, our mechanism demonstrates a rational design principle for engineering information propagation by molecular templating of longer polymers.