  Future actively cooled space-borne observatories for the far-infrared,
loosely defined as a 1--10 THz band, can potentially reach a sensitivity
limited only by background radiation from the Universe. This will result in an
increase in observing speed of many orders of magnitude. A spectroscopic
instrument on such an observatory requires large arrays of detectors with a
sensitivity expressed as a noise equivalent power NEP = 3 $\times 10^{-20}$
$W\surd{Hz}$. We present the design, fabrication, and characterisation of
microwave kinetic inductance detectors (MKIDs) for this frequency range
reaching the required sensitivity. The devices are based on thin-film NbTiN
resonators which use lens-antenna coupling to a submicron-width aluminium
transmission line at the shorted end of the resonator where the radiation is
absorbed. We optimised the MKID geometry for a low NEP by using a small
aluminium volume of $\approx$ 1$\mu m^3$ and fabricating the aluminium section
on a very thin (100 nm) SiN membrane. Both methods of optimisation also reduce
the effect of excess noise by increasing the responsivity of the device, which
is further increased by reducing the parasitic geometrical inductance of the
resonator. We measure the sensitivity of eight MKIDs with respect to the power
absorbed in the detector using a thermal calibration source filtered in a
narrow band around 1.55 THz. We obtain a
NEP$_{exp}(P_{abs})\:=\:3.1\pm0.9\times10^{-20}\:W\surd{Hz}$ at a modulation
frequency of 200 Hz averaged over all measured MKIDs. The NEP is limited by
quasiparticle trapping. The measured sensitivity is sufficient for
spectroscopic observations from future, actively cooled space-based
observatories. Moreover, the presented device design and assembly can be
adapted for frequencies up to $\approx$ 10 THz and can be readily implemented
in kilopixel arrays.
