  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 $Q> 4 \times 10^{15}$.
