Motivated by the ongoing debate about nanophotonic control of Förster resonance energy transfer (FRET), notably by the local density of optical states (LDOS), we study FRET and spontaneous emission in arbitrary nanophotonic media with weak dispersion and weak absorption in the frequency overlap range of donor and acceptor. This system allows us to obtain the following two new insights. Firstly, we derive that the FRET rate only depends on the static part of the Green function. Hence, the FRET rate is independent of frequency, in contrast to spontaneous-emission rates and LDOS that are strongly frequency dependent in nanophotonic media. Therefore, the position-dependent FRET rate and the LDOS at the donor transition frequency are completely uncorrelated for any nondispersive medium. Secondly, we derive an exact expression for the FRET rate as a frequency integral of the imaginary part of the Green function. This leads to very accurate approximation for the FRET rate that features the LDOS that is integrated over a huge bandwidth ranging from zero frequency to far into the UV. We illustrate these general results for the analytic model system of a pair of ideal dipole emitters—donor and acceptor—in the vicinity of an ideal mirror. We find that the FRET rate is independent of the LDOS at the donor emission frequency. Moreover, we observe that the FRET rate hardly depends on the frequency-integrated LDOS. Nevertheless, the FRET is controlled between inhibition and 4×enhancement at distances close to the mirror, typically a few nm. Finally, we discuss the consequences of our results to applications of Förster resonance energy transfer, for instance in quantum information processing.