  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.
