We report the operation of a cold-atom inertial sensor which continuously captures the rotation signal. Using a joint interrogation scheme, where we simultaneously prepare a cold-atom source and operate an atom interferometer (AI) enables us to eliminate the dead times. We show that such continuous operation improves the short-term sensitivity of AIs, and demonstrate a rotation sensitivity of $100 text{nrad.s}^{-1}.text{Hz}^{-1/2}$ in a cold-atom gyroscope of $11 text{cm}^2$ Sagnac area. We also demonstrate a rotation stability of $1 text{nrad.s}^{-1}$ at $10^4$ s of integration time, which establishes the record for atomic gyroscopes. The continuous operation of cold-atom inertial sensors will enable to benefit from the full sensitivity potential of large area AIs, determined by the quantum noise limit.
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