We present an analysis of the mass and age of the young low-mass binary Oph 1622-2405. Using resolved optical spectroscopy of the binary, we measure spectral types of M7.25+/-0.25 and M8.75+/-0.25 for the A and B components, respectively. We show that our spectra are inconsistent with the spectral types of M9 and M9.5-L0 from Jayawardhana & Ivanov and M9+/-0.5 and M9.5+/-0.5 from Close and coworkers. Based on our spectral types and the theoretical evolutionary models of Chabrier and Baraffe, we estimate masses of 0.055 and 0.019 Msun for Oph 1622-2405A and B, which are significantly higher than the values of 0.013 and 0.007 Msun derived by Jayawardhana & Ivanov and above the range of masses observed for extrasolar planets (M<=0.015 Msun). Planet-like mass estimates are further contradicted by our demonstration that Oph 1622-2405A is only slightly later (by 0.5 subclass) than the composite of the young eclipsing binary brown dwarf 2M 0535-0546, whose components have dynamical masses of 0.034 and 0.054 Msun. To constrain the age of Oph 1622-2405, we compare the strengths of gravity-sensitive absorption lines in optical and near-infrared spectra of the primary to lines in field dwarfs (>1 Gyr) and members of Taurus (~1 Myr) and Upper Scorpius (~5 Myr). The line strengths for Oph 1622-2405A are inconsistent with membership in Ophiuchus (<1 Myr) and instead indicate an age similar to that of Upper Sco, which is agreement with a similar analysis performed by Close and coworkers. We conclude that Oph 1622-2405 is part of an older population in Sco-Cen, perhaps Upper Sco itself.
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