  The Fermi National Accelerator Laboratory has measured the anomalous
precession frequency $a^{}_\mu = (g^{}_\mu-2)/2$ of the muon to a combined
precision of 0.46 parts per million with data collected during its first
physics run in 2018. This paper documents the measurement of the magnetic field
in the muon storage ring. The magnetic field is monitored by nuclear magnetic
resonance systems and calibrated in terms of the equivalent proton spin
precession frequency in a spherical water sample at 34.7$^\circ$C. The measured
field is weighted by the muon distribution resulting in $\tilde{\omega}'^{}_p$,
the denominator in the ratio $\omega^{}_a$/$\tilde{\omega}'^{}_p$ that together
with known fundamental constants yields $a^{}_\mu$. The reported uncertainty on
$\tilde{\omega}'^{}_p$ for the Run-1 data set is 114 ppb consisting of
uncertainty contributions from frequency extraction, calibration, mapping,
tracking, and averaging of 56 ppb, and contributions from fast transient fields
of 99 ppb.
