We analyze radial velocity observations of the 12 extremely low-mass \leq 0.25 M _ { \odot } white dwarfs ( WDs ) in the MMT Hypervelocity Star Survey .
Eleven of the 12 WDs are binaries with orbital periods shorter than 14 hours ; 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 can not be excluded .
Six of the 11 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 \leq 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 .