  It has long been expected that quantum degenerate gases of molecules would
open access to a wide range of phenomena in molecular and quantum sciences.
However, the very complexity that makes ultracold molecules so enticing has
made reaching degeneracy an outstanding experimental challenge over the past
decade. We now report the production of a Fermi degenerate gas of ultracold
polar molecules of potassium--rubidium (KRb). Through coherent adiabatic
association in a deeply degenerate mixture of a rubidium Bose-Einstein
condensate and a potassium Fermi gas, we produce molecules at temperatures
below 0.3 times the Fermi temperature. We explore the properties of this
reactive gas and demonstrate how degeneracy suppresses chemical reactions,
making a long-lived degenerate gas of polar molecules a reality.
