  Cold-atom inertial sensors target several applications in navigation,
geoscience and tests of fundamental physics. Reaching high sampling rates and
high inertial sensitivities, obtained with long interrogation times, represents
a challenge for these applications. We report on the interleaved operation of a
cold-atom gyroscope, where 3 atomic clouds are interrogated simultaneously in
an atom interferometer featuring a 3.75 Hz sampling rate and an interrogation
time of 801 ms. Interleaving improves the inertial sensitivity by efficiently
averaging vibration noise, and allows us to perform dynamic rotation
measurements in a so-far unexplored range. We demonstrate a stability of
$3\times 10^{-10}$ rad.s$^{-1}$, which competes with the best stability levels
obtained with fiber-optics gyroscopes. Our work validates interleaving as a key
concept for future atom-interferometry sensors probing time-varying signals, as
in on-board navigation and gravity-gradiometry, searches for dark matter, or
gravitational wave detection.
