It is well known that surface stress can strongly influence the structure and chemical composition of surfaces. Despite significant theoretical progress, few quantitative experimental investigations have been reported. One reason for this is the long-range of elastic interactions at surfaces. To quantitatively determine the influence of surface stress, the structure of the surface over micron-scale distances must be precisely known. In this talk I describe low-energy electron microscopy measurements of Si(111) surface structure near the 7x7 to 1x1 transition temperature (Tc = 1135 K), where phase coexistence is observed. We find that the equilibrium domain geometry is determined by a competition between the free energy difference between the phases, which favors a single phase, and elastic relaxation at the phase boundaries, which favors phase coexistence.@footnote 1@ Elastic relaxation occurs because of the difference in surface stress between the two phases. For Si(111), the stress difference has been measured by Twesten and Gibson.@footnote 2@ In equilibrium, the force on each phase boundary vanishes. We use this fact to determine the free energy difference between phases, and phase boundary creation energy, from the measured domain configurations. In equilibrium, each phase boundary gives rise to an independent equation relating the free energy difference, the boundary creation energy, and the stress difference. By measuring domain configurations as a function of temperature, we determine the temperature dependence of the free energy difference between phases near Tc. Our measurements correspond to an entropy difference between the phases of only 0.01 kB per 1x1 unit cell, a surprisingly small value given the fact that the 7x7 structure has long-range order and the 1x1 phase does not. These results indicate that the degree of disorder in the two phases is similar. @FootnoteText@ @footnote 1@J.B. Hannon, et al., PRL, in press @footnote 2@PRB 50, 17628 (1994).