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Grain boundaries (GBs) in oxide perovskites significantly influence their functional properties. This study examines the atomic-scale structure and composition of a faceted asymmetric grain boundary in strontium titanate (SrTiO$_3$) using scanning transmission electron microscopy (STEM), atom probe tomography (APT), and density functional theory (DFT). STEM and APT reveal an atomically sharp boundary with asymmetric and symmetric facets, marked by strong Sr depletion over a width of less than 1 nm. STEM-EELS shows Ti concentration variations of up to 20% between facets, while APT constrains this variation to less than 10%. DFT$+U$ calculations of a symmetric ${\Sigma}$5 facet confirm that Sr depletion minimizes boundary energy while maintaining Ti content. The variation in Ti suggests cation mobility that enables local energy minimization. Differences in facet surface energies likely drive Ti redistribution, offering strategies for GB structure control. This combined experimental-theoretical approach provides key insights into the structure and energy landscape of GBs in SrTiO$_3$, aiding in the prediction of their impact on ionic and electronic transport.