  We have measured the quadratic Zeeman coefficient for the ${^{1}S_{0}
\leftrightarrow {^{3}P_{0}}}$ optical clock transition in $^{27}$Al$^{+}$,
$C_{2}=-71.944(24)$~MHz/T$^{2}$, and the unperturbed hyperfine splitting of the
$^{25}$Mg$^{+}$ $^{2}S_{1/2}$ ground electronic state, $\Delta W / h =
1~788~762~752.85(13)$~Hz, with improved uncertainties. Both constants are
relevant to the evaluation of the $^{27}$Al$^{+}$ quantum-logic clock
systematic uncertainty. The measurement of $C_{2}$ is in agreement with a
previous measurement and a new calculation at the $1~\sigma$ level. The
measurement of $\Delta W$ is in good agreement with a recent measurement and
differs from a previously published result by approximately $2\sigma$. With the
improved value for $\Delta W$, we deduce an improved value for the
nuclear-to-electronic g-factor ratio $g_{I}/g_{J} = 9.299 ~308 ~313(60) \times
10^{-5}$ and the nuclear g-factor for the $^{25}$Mg nucleus $g_{I} = 1.861 ~957
~82(28) \times 10^{-4}$. Using the values of $C_{2}$ and $\Delta W$ presented
here, we derive a quadratic Zeeman shift of the $^{27}$Al$^{+}$ quantum-logic
clock of $\Delta \nu / \nu = -(9241.8 \pm 3.7) \times 10^{-19}$, for a bias
magnetic field of $B \approx 0.12$~mT.
