We report measurements characterizing the performance of a kinetic inductance detector array designed for a wavelength of 25 microns and very low optical background level suitable for applications such as a far-infrared instrument on a cryogenically cooled space telescope. In a pulse-counting mode of operation at low optical flux, the detectors can resolve individual 25-micron photons. In an integrating mode, the detectors remain photon noise limited over more than 6 orders of magnitude in absorbed power from 70 zW to 200 fW, with a limiting noise equivalent power of 4.6×10-20 W Hz-1 at 1 Hz. In addition, the detectors are highly stable with flat power spectra under optical load down to 1 mHz. Operational parameters of the detector are determined including the efficiency of conversion of the incident optical power into quasiparticles in the aluminum absorbing element and the quasiparticle self-recombination constant.