  Terahertz spectrometers with a wide instantaneous frequency coverage for
passive remote sensing are enormously attractive for many terahertz
applications, such as astronomy, atmospheric science and security. Here we
demonstrate a wide-band terahertz spectrometer based on a single
superconducting chip. The chip consists of an antenna coupled to a transmission
line filterbank, with a microwave kinetic inductance detector behind each
filter. Using frequency division multiplexing, all detectors are read-out
simultaneously creating a wide-band spectrometer with an instantaneous
bandwidth of 45 GHz centered around 350 GHz. The spectrometer has a spectral
resolution of $F/\Delta F=380$ and reaches photon-noise limited sensitivity. We
discuss the chip design and fabrication, as well as the system integration and
testing. We confirm full system operation by the detection of an emission line
spectrum of methanol gas. The proposed concept allows for spectroscopic
radiation detection over large bandwidths and resolutions up to $F/\Delta
F\sim1000$, all using a chip area of a few $\mathrm{cm^2}$. This will allow the
construction of medium resolution imaging spectrometers with unprecedented
speed and sensitivity.
