We analyze the spectrum of an unusually low mass white dwarf, SDSS J123410.37-022802.9 (0335-264-52000), found in our recent, white dwarf catalog from the first data release (DR1) of the Sloan Digital Sky Survey (SDSS). Two independent, model atmosphere fits result in an accurate determination of atmospheric and stellar parameters. The more hands-on analysis yields Teff=17,470+/-750 K and logg=6.38+/-0.05. We argue that the object cannot be a main-sequence A star, a horizontal-branch star, or a subdwarf B star. Instead, it is interpreted as a very low mass white dwarf with a core composed of helium, with mass ~0.18-0.19 Msolar, similar to that published previously for the unusual companion to the millisecond pulsar J1012+5307. The star probably remains in a binary, perhaps even with an undiscovered or dead pulsar companion. However, the companion might be a more ordinary star, provided Roche lobe mass transfer began shortly after the now-visible component left the main sequence. A second SDSS low-mass white dwarf candidate is also analyzed, but the spectrum for this fainter object is of poorer quality. The sample appears to include additional, similar candidates, worthy of more accurate observation. Correct identification of any additional white dwarfs of extremely low mass requires careful observation and interpretation.