  We present measurements of the hyperfine structure splittings of $nS_{1/2}$
Rydberg states of $^{85}$Rb for $n$= 43, 44, 45 and 46. From the splittings,
the hyperfine coupling constant, $A_{\mathrm{HFS}}$, is determined to be
15.372(80) GHz. This result is an order-of-magnitude improvement from previous
measurements. We study and account for systematic uncertainty sources, such as
unwanted electric and magnetic fields, dipolar Rydberg-Rydberg interactions,
and AC shifts. Initial evidence for hyperfine-mixed Rydberg pair states is
found.
