  We utilize two-photon high-precision microwave spectroscopy of
$ng\rightarrow(n+2)g$ transitions to precisely measure the
high-angular-momentum $g$-series quantum defect of $^{85}$Rb. Samples of cold
Rydberg atoms in the $ng$ state are prepared via a three-photon optical
excitation combined with controlled electric-field mixing and probed with
40-$\mu$s-long microwave interaction pulses. The leading systematic uncertainty
arises from DC Stark shifts, which is addressed by a cancellation of background
electric fields in all three dimensions. From our measurements and an analysis
of systematic uncertainties from DC and AC Stark shifts, van der Waals
interactions, and microwave frequency calibration, we obtain
$\delta_0=0.0039990(21)$ and $\delta_2=-0.0202(21)$. We discuss our results in
context with recent work elsewhere, as well as applications towards precision
measurement.
