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We present Far-Ultraviolet Spectroscopic Explorer observations of the atomic and molecular absorption in high-velocity cloud HVC 287.5+22.5+240, which lies in front of the ultraviolet-bright nucleus of the Seyfert 1 galaxy NGC 3783. We detect H₂, N I, N II, Si II, and Fe II and set limits on the amount of absorption due to P III, Ar I, and Fe III. We extend the earlier metallicity and dust-depletion measurements made by Lu and collaborators by examining the relative gas-phase abundances of Si, P, S, and Fe. Corrections to the derived gas-phase abundances due to ionized gas in the HVC are small (≲15%). The HVC has metallicity 0.2-0.4 solar, similar to that of the Small Magellanic Cloud. The relative abundance pattern for the elements studied resembles that of warm gas in the Small Magellanic Cloud (SMC), which supports the idea that this HVC is part of the tidally stripped leading arm of the Magellanic Stream. The abundance pattern implies that the HVC contains dust grains that have been processed significantly; it is likely that the grain mantles have been modified or stripped back to expose the grain cores. We have identified more than 30 lines of H₂ arising in the HVC from rotational levels J=0 to J=3. Synthetic spectra and a curve of growth fitted to these lines with b=12 km.s^–1 indicate that log[N(H₂)]=16.80±0.10 and f_H2=2N(H₂)/[N(H I)+2N(H₂)]=1.6x10^–3. A two-component temperature distribution is necessary to explain the observed populations of the H₂ rotational levels. We find T₀₁=133⁺³⁷_–21 K, and T₂₃=241⁺²⁰_–17 K, indicating that the conditions in the molecular gas are more similar to those found for diffuse molecular clouds in the Galactic halo than to those for molecular clouds in the Galactic disk. From an analysis of the J=2 and J=3 populations, we find an absorption rate (at 1000 Å) of β_uv<0.1 times the average value in the solar neighborhood. The presence of molecular gas in the HVC requires that either the H₂ formed in situ or that molecules formed within the SMC survived tidal stripping. We favor the latter possibility because of the long H₂ formation time (∼10⁸ yr) derived for this HVC.