Measurements of $D^{*}(2010)$ meson production in diffractive deep inelastic scattering $ (5 < Q^{2} < 100~{\rm GeV}^{2}) $ are presented which are based on HERA data recorded at a centre-of-mass energy $\sqrt{s} = 319{\rm~GeV}$ with an integrated luminosity of $287$ pb$^{-1}$ collected by the H1 detector. The reaction $ep \rightarrow eXY$ is studied, where the system $X$, containing at least one $D^{*}(2010)$ meson, is separated from a leading low-mass proton dissociative system $Y$ by a large rapidity gap. The kinematics of $D^{*}$ candidates are reconstructed in the $D^{*}\rightarrow K \pi\pi$ decay channel. The measured cross sections compare favourably with next-to-leading order QCD predictions, where charm quarks are produced via boson-gluon fusion and they are independently fragmented to the $D^{*}$ mesons. The calculations rely on the collinear factorisation theorem and are based on diffractive parton densities previously obtained by H1 from fits to inclusive diffractive cross sections. The data are further used to determine the diffractive to inclusive $D^{*}$ production ratio in deep inelastic scattering.