We present high angular resolution observations of the Class 0 protostar IRAM04191+1522, using the Submillimeter Array (SMA). The SMA 1.3 mm continuum images reveal within IRAM04191+1522 two distinct sources with an angular separation of 7.8,$pm$,0.2$$. The two continuum sources are located in the southeast-northwest direction, with total gas masses of about 0.011 M_sun and about 0.005 M_sun, respectively. The southeastern source, associated with an infrared source seen in the Spitzer images, is the well-known Class 0 protostar with a bolometric luminosity of about 0.08 L_sun. The newly-discovered northwestern continuum source is not visible in the Spitzer images at wavelengths from 3.6 to 70 micron, and has an extremely low bolometric luminosity (< 0.03 L_sun). Complementary IRAM N2H+(1-0) data that probe the dense gas in the common envelope suggest that the two sources were formed through the rotational fragmentation of an elongated dense core. Furthermore, comparisons between IRAM04191+1522 and other protostars suggest that most cores with binary systems formed therein have ratios of rotational energy to gravitational energy $beta_{rm rot}$ > 1%. This is consistent with theoretical simulations and indicates that the level of rotational energy in a dense core plays an important role in the fragmentation process.
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