We present high angular resolution observations of the Class 0 protostar IRAM 04191+1522 using the Submillimeter Array (SMA). The SMA 1.3 mm continuum images reveal within IRAM 04191+1522 two distinct sources with an angular separation of 7".8±0".2. The two continuum sources are located in the southeast-northwest direction, with total gas masses of ∼0.011 M☉ and ∼0.005 M☉, 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 ∼0.08 L☉. The newly discovered northwestern continuum source is not visible in the Spitzer images at wavelengths from 3.6 to 70 µm and has an extremely low bolometric luminosity (<0.03 L☉). 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 IRAM 04191+1522 and other protostars suggest that most cores with binary systems formed therein have ratios of rotational energy to gravitational energy βrot> 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.