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The principle of a terahertz detector using a superconducting tunnel junction (STJ) coupled to a large terahertz-absorbing superconductor film is verified. We have detected terahertz radiation based on the Cooper-pair breaking process, and confirmed that the sensitivity has a sharp increase around 0.7 THz, a value that is in agreement with the gap frequency of the superconducting Nb. For high-sensitivity terahertz detection, we propose an improved design of the STJ detector composed of a smaller junction coupled to a thinner superconductor film. We also discuss the expected noise equivalent power of the optimized detector.