Atomic clocks based on optical transitions are the most stable, and therefore precise, timekeepers available. These clocks operate by alternating intervals of atomic interrogation with dead time required for quantum state preparation and readout. This non-continuous interrogation of the atom system results in the Dick effect, an aliasing of frequency noise of the laser interrogating the atomic transition. Despite recent advances in optical clock stability achieved by improving laser coherence, the Dick effect has continually limited optical clock performance. Here we implement a robust solution to overcome this limitation: a zero-dead-time optical clock based on the interleaved interrogation of two cold-atom ensembles. This clock exhibits vanishingly small Dick noise, thereby achieving an unprecedented fractional frequency instability of $6 times 10^{-17} / sqrt{tau}$ for an averaging time $tau$ in seconds. We also consider alternate dual-atom-ensemble schemes to extend laser coherence and reduce the standard quantum limit of clock stability, achieving a spectroscopy line quality factor $Q> 4 times 10^{15}$.
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