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We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$${\rho }^0$meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$${\mu }^{+}$and$$ \mu ^{-}$$${\mu }^{-}$beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$$c^2$$$< W<$$$<W<$17.0 GeV/$$c^2$$$c^2$, 1.0 (GeV/c)$$^2$$${}^2$$$< Q^2<$$$<Q^2<$10.0 (GeV/c)$$^2$$${}^2$and 0.01 (GeV/c)$$^2$$${}^2$$$< p_{\textrm{T}}^2<$$$<p_{\text{T}}^2<$0.5 (GeV/c)$$^2$$${}^2$. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$$Q^2$the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$$p_{\text{T}}$the transverse momentum of the$$\rho ^0$$${\rho }^0$meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$${\gamma }_T^{\ast }\to V_L^{}$) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$${\rho }^0$production.