  A method for performing a precision measurement of the Rydberg constant,
$R_{\infty}$, using cold circular Rydberg atoms is proposed. These states have
long lifetimes, as well as negligible quantum-electrodynamics (QED) and no
nuclear-overlap corrections. Due to these advantages, the measurement can help
solve the "proton radius puzzle" [Bernauer, Pohl, Sci. Am. 310, 32 (2014)]. The
atoms are trapped using a Rydberg-atom optical lattice, and transitions are
driven using a recently-demonstrated lattice-modulation technique to perform
Doppler-free spectroscopy. The circular-state transition frequency yields
$R_{\infty}$. Laser wavelengths and beam geometries are selected such that the
lattice-induced transition shift is minimized. The selected transitions have no
first-order Zeeman and Stark corrections, leaving only manageable second-order
Zeeman and Stark shifts. For Rb, the projected relative uncertainty of
$R_{\infty}$ in a measurement under the presence of the Earth's gravity is
$10^{-11}$, with the main contribution coming from the residual lattice shift.
This could be reduced in a future micro-gravity implementation. The
next-important systematic arises from the Rb$^+$ polarizability
(relative-uncertainty contribution of $\approx 3 \times10^{-12}$).
