  We report precision measurements of the nuclear magnetic moment of
\textsuperscript{43}Ca\textsuperscript{+}, made by microwave spectroscopy of
the 4s $^2$S$_{1/2}$ $\left|F=4, M=0\right\rangle \rightarrow \left|F=3,
M=1\right\rangle$ ground level hyperfine clock transition at a magnetic field
of $\approx$ 146 G, using a single laser-cooled ion in a Paul trap. We measure
a clock transition frequency of $f = 3199941076.920 \pm 0.046$ Hz, from which
we determine $\mu_I / \mu_{\rm{N}} = -1.315350(9)(1)$, where the uncertainty
(9) arises from uncertainty in the hyperfine $A$ constant, and the (1) arises
from the uncertainty in our measurement. This measurement is not corrected for
diamagnetic shielding due to the bound electrons. We make a second measurement
which is less precise but agrees with the first. We use our $\mu_I$ value, in
combination with previous NMR results, to extract the change in shielding
constant of calcium ions due to solvation in D$_2$O: $\Delta \sigma =
-0.00022(1)$.
