  We report precise measurement of the hyperfine splitting and calculation of
the Zeeman coefficients of the $^{171}$Yb$^+$ ground state. The absolute
hyperfine splitting frequency is measured using high-resolution laser-microwave
double-resonance spectroscopy at 0.1 mHz level, and evaluated using more
accurate Zeeman coefficients. These Zeeman coefficients are derived using
Land\'{e} $g_J$ factors calculated by two atomic-structure methods,
multiconfiguration Dirac-Hartree-Fock, and multireference configuration
interaction. The cross-check of the two calculations ensures an accuracy of the
Zeeman coefficients at $10^{-2}$ Hz/G$^2$ level. The results provided in this
paper improve the accuracy and reliability of the second-order Zeeman shift
correction, thus further improving the accuracy of the microwave frequency
standards based on $^{171}$Yb$^+$. The high-precision hyperfine splitting and
Zeeman coefficients could also support could also support further experiments
to improve the constraints of fundamental constants through clock frequency
comparison of the Yb$^+$ system.
