  We perform a comprehensive investigation of F\"orster resonant energy
transfer in a room-temperature thermal mixture of ammonia molecules and
rubidium Rydberg atoms. Fully state-resolved measurement of the Rydberg-atom
populations is achieved by combining millimeter-wave state transfer with
state-selective field ionization. This allows aspects of the energy transfer
process such as state dependence, ammonia pressure dependence, and dependence
on the energy resonance condition to be investigated in detail. Our results
pave the way for future quantum experiments combining polar molecules and
Rydberg atoms.
