  We describe an optical atomic clock based on quantum-logic spectroscopy of
the $^1$S$_0$ $\leftrightarrow$ $^3$P$_0$ transition in $^{27}$Al$^{+}$ with a
systematic uncertainty of ${9.4 \times 10^{-19}}$ and a frequency stability of
${1.2\times10^{-15}/\sqrt{\tau}}$. A $^{25}$Mg$^{+}$ ion is simultaneously
trapped with the $^{27}$Al$^{+}$ ion and used for sympathetic cooling and state
readout. Improvements in a new trap have led to reduced secular motion heating,
compared to previous $^{27}$Al$^{+}$ clocks, enabling clock operation with ion
secular motion near the three-dimensional ground state. Operating the clock
with a lower trap drive frequency has reduced excess micromotion compared to
previous $^{27}$Al$^{+}$ clocks. Both of these improvements have led to a
reduced time-dilation shift uncertainty. Other systematic uncertainties
including those due to blackbody radiation and the second-order Zeeman effect
have also been reduced.
