We present radial velocity observations of four extremely low-mass (0.2 Msol) white dwarfs. All four stars show peak-to-peak radial velocity variations of 540 - 710 km/s with 1.0 - 5.9 hr periods. The optical photometry rules out main-sequence companions. In addition, no milli-second pulsar companions are detected in radio observations. Thus the invisible companions are most likely white dwarfs. Two of the systems are the shortest period binary white dwarfs yet discovered. Due to the loss of angular momentum through gravitational radiation, three of the systems will merge within 500 Myr. The remaining system will merge within a Hubble time. The mass functions for three of the systems imply companions more massive than 0.46 Msol; thus those are carbon/oxygen core white dwarfs. The unknown inclination angles prohibit a definitive conclusion about the future of these systems. However, the chance of a supernova Ia event is only 1% to 5%. These systems are likely to form single R Coronae Borealis stars, providing evidence for a white dwarf + white dwarf merger mechanism for these unusual objects. One of the systems, SDSS J105353.89+520031.0 has a 70% chance of having a low-mass white dwarf companion. This system will probably form a single helium-enriched subdwarf O star. All four white dwarf systems have unusal mass ratios of < 0.2-0.8 that may also lead to the formation of AM CVn systems.Comment: ApJ, accepted versio
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