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We report a measurement of the hyperfine-structure constants of the⁸⁵Rb 4$D_{3/2}$state using two-photon optical spectroscopy of the 5$S_{1/2}\to$4$D_{3/2}$transition. The spectra are acquired by measuring the transmission of the low-power 795 nm lower-stage laser beam through a cold-atom sample as a function of laser frequency, with the frequency of the upper-stage, 1476 nm laser fixed. All 4 hyperfine components of the$4D_{3/2}$state are well-resolved in the experimental data. The dominant systematic is the light shift from the 1476 nm laser, which is addressed by extrapolating line positions measured for a set of 1476 nm laser powers to zero laser power. The analysis of our experimental data yields both the magnetic-dipole and electric-quadrupole constants for the⁸⁵Rb 4$D_{3/2}$level, without using earlier hyperfine measurements of other atomic levels. The respective results,$A=$7.419(35) MHz and$B=$4.19(19) MHz, are discussed in context with previous works. Our investigation may be useful for optical atomic clocks for precision metrology and emerging atom-based quantum technologies, all-infrared excitation of Rb Rydberg levels, and molecular physics.