  We present a method to precisly measure the frequencies of transitions to
high-$n$ Rydberg states of the hydrogen atom which are not subject to
uncontrolled systematic shifts caused by stray electric fields. The method
consists in recording Stark spectra of the field-insensitive $k=0$ Stark states
and the field-sensitive $k=\pm2$ Stark states, which are used to calibrate the
electric field strength. We illustrate this method with measurements of
transitions from the $2\,\text{s}(f=0\text{ and } 1)$ hyperfine levels in the
presence of intentionally applied electric fields with strengths in the range
between $0.4$ and $1.6\,$Vcm$^{-1}$. The slightly field-dependent $k=0$ level
energies are corrected with a precisely calculated shift to obtain the
corresponding Bohr energies $\left(-cR_{\mathrm{H}}/n^2\right)$. The energy
difference between $n=20$ and $n=24$ obtained with our method agrees with
Bohr's formula within the $10\,$kHz experimental uncertainty. We also
determined the hyperfine splitting of the $2\,\text{s}$ state by taking the
difference between transition frequencies from the $2\,\text{s}(f=0 \text{ and
}1)$ levels to the $n=20,k=0$ Stark states. Our results demonstrate the
possibility of carrying out precision measurements in high-$n$ hydrogenic
quantum states.
