Ramified silver aggregates were obtained in thin-gap cells using a simple displacement reaction. No external field was applied and convection was eliminated by reducing the gap size, so that this electroless growth can be examined in terms of a diffusion-limited A + B --> C reaction. After a detailed examination of some experimental problems, the final morphology of the aggregates is briefly compared with the diffusion-limited aggregation (DLA) morphology of the Witten-Sander model. The spatio-temporal evolution of the morphology is then compared with the predictions of a reaction-diffusion model and good agreement is found for the universal time exponents alpha and beta of the motion of the boundary zone and the rate of global product formation respectively.