We analyze the spectrum of an unusually low mass white dwarf, SDSSJ123410.37-022802.9 (0335-264-52000), found in our recent, white dwarfcatalog from the first data release (DR1) of the Sloan Digital SkySurvey (SDSS). Two independent, model atmosphere fits result in anaccurate determination of atmospheric and stellar parameters. The morehands-on analysis yields Teff=17,470+/-750 K andlogg=6.38+/-0.05. We argue that the object cannot be a main-sequence Astar, a horizontal-branch star, or a subdwarf B star. Instead, it isinterpreted as a very low mass white dwarf with a core composed ofhelium, with mass ~0.18-0.19 Msolar, similar to thatpublished previously for the unusual companion to the millisecond pulsarJ1012+5307. The star probably remains in a binary, perhaps even with anundiscovered or dead pulsar companion. However, the companion might be amore ordinary star, provided Roche lobe mass transfer began shortlyafter the now-visible component left the main sequence. A second SDSSlow-mass white dwarf candidate is also analyzed, but the spectrum forthis fainter object is of poorer quality. The sample appears to includeadditional, similar candidates, worthy of more accurate observation.Correct identification of any additional white dwarfs of extremely lowmass requires careful observation and interpretation.