  Helium-3 has nowadays become one of the most important candidates for studies
in fundamental physics [1, 2, 3], nuclear and atomic structure [4, 5],
magnetometry and metrology [6] as well as chemistry and medicine [7, 8]. In
particular, $^3$He nuclear magnetic resonance (NMR) probes have been proposed
as a new standard for absolute magnetometry [6, 9]. This requires a
high-accuracy value for the $^3$He nuclear magnetic moment, which, however, has
so far been determined only indirectly and with a relative precision of $12$
parts per billon (p.p.b.) [10,11]. Here we investigate the $^3$He$^+$
ground-state hyperfine structure in a Penning trap to directly measure the
nuclear $g$-factor of $^3$He$^+$ $g'_I=-4.255\, 099\, 606\,
9(30)_{stat}(17)_{sys}$, the zero-field hyperfine splitting $E_{\rm HFS}^{\rm
exp}=-8\, 665\, 649\, 865.77(26)_{stat}(1)_{sys}$ Hz and the bound electron
$g$-factor $g_e^\text{exp}=-2.002\, 177\, 415\, 79(34)_{stat}(30)_{sys}$. The
latter is consistent with our theoretical value $g_e^\text{theo}=-2.002\, 177\,
416\, 252\, 23(39)$ based on parameters and fundamental constants from [12].
Our measured value for the $^3$He$^+$ nuclear $g$-factor allows for the
determination of the $g$-factor of the bare nucleus $g_I=-4.255\, 250\, 699\,
7(30)_{stat}(17)_{sys}(1)_{theo}$ via our accurate calculation of the
diamagnetic shielding constant [13] $\sigma_{^3He^+}=0.000\,035\,507\,38(3)$.
This constitutes the first direct calibration for $^3$He NMR probes and an
improvement of the precision by one order of magnitude compared to previous
indirect results. The measured zero-field hyperfine splitting improves the
precision by two orders of magnitude compared to the previous most precise
value [14] and enables us to determine the Zemach radius [15] to
$r_Z=2.608(24)$ fm.
