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Combining visual and spectroscopic orbits of binary stars leads to a determination of the full 3D orbit, individual masses, and distance to the system. We present a full analysis of the evolved binary system δ Delphini using astrometric data from the MIRC and PAVO instruments on the CHARA long-baseline interferometer, 97 new spectra from the Fairborn Observatory, and 87 unpublished spectra from the Lick Observatory. We determine the full set of orbital elements for δ Del, along with masses of 1.78 ± 0.07 M⊙ and 1.62 ± 0.07 M⊙ for each component, and a distance of 63.61 ± 0.89 pc. These results are important in two contexts: for testing stellar evolution models and for defining the detection capabilities for future planet searches. We find that the evolutionary state of this system is puzzling, as our measured flux ratios, radii, and masses imply a ∼200 Myr age difference between the components, using standard stellar evolution models. Possible explanations for this age discrepancy include mass transfer scenarios with a now-ejected tertiary companion. For individual measurements taken over a span of two years, we achieve <10 μas precision on the differential position with 10 minute observations. The high precision of our astrometric orbit suggests that exoplanet detection capabilities are within reach of MIRC at CHARA. We compute exoplanet detection limits around δ Del and conclude that, if this precision is extended to wider systems, we should be able to detect most exoplanets >2 MJ on orbits >0.75 au around individual components of hot binary stars via differential astrometry.