We report the operation of a cold-atom inertial sensor in a joint interrogation scheme, where we simultaneously prepare a cold-atom source and operate an atom interferometer in order to eliminate dead times. Noise aliasing and dead times are consequences of the sequential operation which is intrinsic to cold-atom atom interferometers. Both phenomena have deleterious effects on the performance of these sensors. We show that our continuous operation improves the short-term sensitivity of atom interferometers, by demonstrating a record rotation sensitivity of $100$ nrad.s$^{-1}/sqrt{rm Hz}$ in a cold-atom gyroscope of $11$ cm$^2$ Sagnac area. We also demonstrate a rotation stability of $1$ nrad.s$^{-1}$ after $10^4$ s of integration, improving previous results by an order of magnitude. We expect that the continuous operation will allow cold-atom inertial sensors with long interrogation time to reach their full sensitivity, determined by the quantum noise limit.
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