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/c polarised $$ \mu ^{+}$$ ${\mu }^{+}$ 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, W denotes 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 of s-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.