We show that the coupling responsible for fluorescence resonance energy transfer (FRET) generates a mechanical force that is distinct from the van der Waals force between ground-state molecules and can be either attractive or repulsive. The underlying potential has the same 1/r(6) distance dependence as the rate of FRET, and the two are connected by a Kramers-Kronig relation. Just as the rate of FRET can be derived either classically or quantum mechanically, so too can the interaction potential. Because of the FRET force, time-resolved FRET measurements contain information on the mechanical stiffness of the matrix containing donor and acceptor.