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Brown dwarfs with known physical properties (e.g., age and mass) are essential for constraining models of the formation and evolution of substellar objects. We present new high-contrast imaging observations of the circumbinary brown dwarf H II 1348B-one of the few known substellar companions in the Pleiades cluster. We observed the system in the infrared $L^{\prime} $ band with the Large Binocular Telescope Interferometer in dual-aperture direct-imaging mode (i.e., with the two telescope apertures used separately) on 2019 September 18. The observations attained a high signal-to-noise ratio (SNR > 150) photometric detection and relative astrometry with uncertainties of ∼5 mas. This work presents the first model of the companion's orbital motion using relative astrometry from five epochs across a total baseline of 23 yr. Orbital fits to the compiled data show the companion's semimajor axis to be $a=14{0}_{-30}^{+130}\,{\rm{au}}$ with an eccentricity of $e=0.7{8}_{-0.29}^{+0.12}$. We infer that H II 1348B has a mass of 60-63 ± 2 M_J from comparison to brown dwarf evolutionary models given the well-constrained distance and age of the Pleiades. No other objects were detected in the H II 1348 system, and through synthetic planet injection and retrievals we establish detection limits at a cluster age of 112 ± 5 Myr down to ∼10-30 M_J for companions with projected separations of 21.5-280 au. With this work, H II 1348B becomes the second directly imaged substellar companion in the Pleiades with measured orbital motion after HD 23514B.