During a survey for stars with disks in the Taurus star-forming region using the Spitzer Space Telescope, we have discovered a pair of young brown dwarfs, FU Tau A and B, in the Barnard 215 dark cloud. They have a projected angular separation of 5.7", corresponding to 800 AU at the distance of Taurus. To assess the nature of these two objects, we have obtained spectra of them and have constructed their spectral energy distributions. Both sources are young (~1 Myr) according to their Halpha emission, gravity-sensitive spectral features, and mid-IR excess emission. The proper motion of FU Tau A provides additional evidence of its membership in Taurus. We measure spectral types of M7.25 and M9.25 for FU Tau A and B, respectively, which correspond to masses of ~0.05 and ~0.015 M\cdot according to the evolutionary models of Chabrier and Baraffe. FU Tau A is significantly overluminous relative to an isochrone passing through FU Tau B and relative to other members of Taurus near its spectral type, which may indicate that it is an unresolved binary. FU Tau A and B are likely to be components of a binary system based on the low probability (~3x10^-4) that Taurus would produce two unrelated brown dwarfs with a projected separation of a </- 6". Barnard 215 contains only one other young star and is in a remote area of Taurus, making FU Tau A and B the first spectroscopically-confirmed brown dwarfs discovered forming in isolation rather than in a stellar cluster or aggregate. Because they were born in isolation and comprise a weakly bound binary, dynamical interactions with stars could not have played a role in their formation, and thus are not essential for the birth of brown dwarfs. ERRATUM: The K-band magnitude for FU Tau B in Table 1 is incorrect and should be 13.33. The bolometric luminosity of FU Tau B in Table 3 and Figure 5 is incorrect because of that mistake and a separate arithmetic error. The correct value of the luminosity is 0.0039 Lsun. FU Tau A and B exhibited different isochronal ages in the original Hertzsprung-Russell diagram in Figure 5, which was unexpected for members of a binary system. This discrepancy is reduced in the corrected version of Figure 5 since both objects are now above the isochrone for 1 Myr. Given the large uncertainties in model isochrones at such young ages, the positions of FU Tau A and B in Figure 5 could be roughly consistent with coevality.
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