We analyze radial velocity observations of the 12 extremely low-mass (ELM), with <=0.25 M sun, white dwarfs (WDs) in the MMT Hypervelocity Star Survey. Eleven of the twelve WDs are binaries with orbital periods shorter than 14 hr; the one non-variable WD is possibly a pole-on system among our non-kinematically selected targets. Our sample is unique: it is complete in a well-defined range of apparent magnitude and color. The orbital mass functions imply that the unseen companions are most likely other WDs, although neutron star companions cannot be excluded. Six of the eleven systems with orbital solutions will merge within a Hubble time due to the loss of angular momentum through gravitational wave radiation. The quickest merger is J0923+3028, a g = 15.7 ELM WD binary with a 1.08 hr orbital period and a <=130 Myr merger time. The chance of a supernova Ia event among our ELM WDs is only 1%-7%, however. Three binary systems (J0755+4906, J1233+1602, and J2119-0018) have extreme mass ratios and will most likely form stable mass-transfer AM CVn systems. Two of these objects, SDSS J1233+1602 and J2119-0018, are the lowest surface gravity WDs ever found; both show Ca II absorption likely from accretion of circumbinary material. We predict that at least one of our WDs is an eclipsing detached double WD system, important for constraining helium core WD models. Based on observations obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona.