Good question! The coldest (and driest) desert in the world is the McMurdo Dry Valleys area of Antarctica. One study of the climate of the Dry Valleys from 1986–2000 showed that the average annual temperature of different locations in the Dry Valley ranged from approximately –15 to –30 Celsius (5 to –22 Fahrenheit; Doran et al., 2002). That is well below freezing!

references
Doran, P.T. et al., 2002. Valley floor climate observations from the McMurdo dry valleys, Antarctica, 1986¬–2000. Journal of Geophysical Research: Atmospheres, 107, 13-1–13-12.