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Using new radial velocities obtained at KPNO and Fairborn Observatory, we have determined improved spectroscopic orbits for two double-lined F-type binaries, HD 24623 and V923 Sco. The orbital periods are 19.66304 and 34.8386 days, respectively, so it is not surprising that their orbits have relatively high eccentricities of nearly 0.5. The orbital dimensions (a1sin i and a2sin i) and minimum masses (m1sin 3i and m2sin 3i) have accuracies of 0.2% or better. Extensive photometry of HD 24623 with the T4 0.75 m automatic photometric telescope at Fairborn Observatory shows no evidence of eclipses. Instead, a very weak reflection effect is seen, making the system a new variable star. Our spectroscopic ephemeris for V923 Sco indicates that the eclipse detected by Bolton and Herbst is a partial eclipse of the primary but detection of the secondary eclipse is uncertain. For HD 24623, we have determined spectral types of F2 dwarf and F4 dwarf for the primary and secondary, respectively. Our spectral types are F4 dwarf for the two components of V923 Sco. Both components of HD 24623 are rotating more slowly than their pseudosynchronous velocities, as is the primary of V923 Sco. However, the secondary of V923 Sco is likely rotating pseudosynchronously.