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A new SQUID based measurement system for detecting dark currents, generated by the TESLA (Tera Electron Volt Energy Superconducting Linear Accelerator) cavities is proposed. It makes use of the Cryogenic Current Comparator principle and senses dark currents in the nA range. To reach the maximum possible energy in the TESLA project is a strong motivation to push the gradients of the superconducting cavities closer to the physical limit of 50 MV/m. The field emission of electrons (dark current) of the cavities at strong fields may limit the maximum gradient. The absolute measurement of the dark current in correlation with the gradient will give a proper value to compare and classify the cavities. This contribution describes a Cryogenic Current Comparator (CCC) as an excellent tool for this purpose. The most important component of the CCC is a high performance DC SQUID system which is able to measure extremely low magnetic fields, e.g., caused by the extracted dark current. For this reason the SQUID input coil is connected across a special designed pick-up coil for the electron beam. Both the SQUID input coil and the pick-up coil form a closed superconducting loop so that the CCC is able to detect dc currents down to 1 nA/√Hz. Design issues and the application for the CHECHIA (horizontal test cryostat) cavity test stand at DESY are discussed.