For diffusion-controlled reactions in three dimensions, continuum mechanics provides a quantitative relation between the steady-state reaction rate constant k and the diffusion coefficient D. However, this approach fails in two dimensions, where no steady-state solution exists on an infinite domain. Using both Monte Carlo methods and analytical techniques, we show that previous attempts to circumvent this problem fail under real laboratory conditions, where fractional coverages often exceed 10(-3). Instead, we have developed a rigorous and general relation between k and D for all coverages on a square lattice for the reaction A + A --> A(2). For short times or high coverages, the relation k = pi D/gamma holds exactly where gamma denotes the two-dimensional packing fraction. For lower coverages, however, k depends on time in both constant-coverage (adsorption allowed) and transient-coverage (adsorption forbidden) regimes. In both cases, k decreases in response to the evolution of nonrandom adsorbate configurations on the surface. These results indicate that diffusion-limited surface reactions may be identified unambiguously in the laboratory and also provide a quantitative link between diffusion parameters and experimentally determined recombination rate parameters. Practical experimental methods highlighting such effects are outlined.