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Young stars surrounded by disks with very low mass accretion rates are likely in the final stages of inner disk evolution and therefore particularly interesting to study. We present ultraviolet (UV) observations of the ~5-9 Myr old stars RECX-1 and RECX-11, obtained with the Cosmic Origins Spectrograph and Space Telescope Imaging Spectrograph on the Hubble Space Telescope, as well as optical and near-infrared spectroscopic observations. The two stars have similar levels of near-UV emission, although spectroscopic evidence indicates that RECX-11 is accreting and RECX-1 is not. The line profiles of Hα and He I λ10830 in RECX-11 show both broad and narrow redshifted absorption components that vary with time, revealing the complexity of the accretion flows. We show that accretion indicators commonly used to measure mass accretion rates, e.g., U-band excess luminosity or the Ca II triplet line luminosity, are unreliable for low accretors, at least in the middle K spectral range. Using RECX-1 as a template for the intrinsic level of photospheric and chromospheric emission, we determine an upper limit of 3 × 10^(–10) M_☉ yr–1 for RECX-11. At this low accretion rate, recent photoevaporation models predict that an inner hole should have developed in the disk. However, the spectral energy distribution of RECX-11 shows fluxes comparable to the median of Taurus in the near-infrared, indicating that substantial dust remains. Fluorescent H2 emission lines formed in the innermost disk are observed in RECX-11, showing that gas is present in the inner disk, along with the dust.