The census of young moving groups in the solar neighborhood is significantly incomplete in the low-mass regime. We have developed a new selection process to find these missing members based on the GALEX All-Sky Imaging Survey. For stars with spectral types >K5 and younger than 300~Myr, we show that near-UV and far-UV emission is greatly enhanced above the quiescent photosphere, analogous to the enhanced X-ray emission of young low-mass stars seen by ROSAT but detectable to much larger distances with GALEX. By combining GALEX data with optical (HST Guide Star Catalog) and near-IR (2MASS) photometry, we identified an initial sample of 34 young M dwarf candidates in a 1000 sq.~deg.~region around the 10-Myr TW Hydra Association (TWA). Low-resolution spectroscopy of 30 of these found 16 which had H_alpha in emission, which were then followed-up at high resolution to search for spectroscopic evidence of youth and to measure radial velocities. Four objects have low surface gravities, photometric distances and space motions consistent with TWA, but the non-detection of Li indicates they may be too old to belong to this moving group. One object (M3.5, 93 pc) appears to be the first known accreting low-mass member of the 15~Myr Lower Centaurus Crux OB association. Two objects exhibit all the characteristics of the known TWA members, and thus we designate them as TWA 31 (M4.2, 110 pc) and TWA 32 (M6.3, 53 pc). TWA 31 shows extremely broad (447 km/s) H_alpha emission, making it the sixth member of TWA found to have ongoing accretion. TWA 32 is resolved into a 0.6" binary in Keck laser guide star adaptive optics imaging. Our search should be sensitive down to spectral types of at least M4-M5 in TWA and thus the small numbers of new member is puzzling. This may indicate TWA has an atypical mass function or that the presence of Li may be too restrictive a criteria for selecting young low-mass stars.
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