Aims. 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 × 10-20 W√ p Hz. Methods. 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 ≈1 μm3 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. Results. 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 1.5 THz. We obtain a NEPexp(Pabs) = 3:1 ± 0:9 × 10-20 W√ p Hz at a modulation frequency of 200 Hz averaged over all measured MKIDs. The NEP is limited by quasiparticle trapping. Conclusions. 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 ≈10 THz and can be readily implemented in kilopixel arrays.