  We report a measurement of the hyperfine-structure constants of the $^{85}$Rb
4$D_{3/2}$ state using a two-photon 5$S_{1/2}\rightarrow$4$D_{3/2}$ transition.
The hyperfine transitions are probed by measuring the transmission of the
low-power 795-nm lower-stage laser beam through a cold-atom sample as a
function of 795-nm laser frequency, with the frequency of the upper-stage
1476-nm laser fixed. All 4 hyperfine components are well-resolved in the
recorded transmission spectra. AC shifts are carefully considered. The
field-free hyperfine line positions are obtained by extrapolating measured line
positions to zero laser power. The magnetic-dipole and electric-quadrupole
constants, $A$ and $B$, are determined from the hyperfine intervals to be
7.419(35)~MHz and 4.19(19)~MHz, respectively. The results are evaluated in
context with previous works. Possible uses of the Rb 4$D_J$ states in
Rydberg-atom-physics, precision-metrology and quantum-technology applications
are discussed.
