  High-pressure magnetization, structural and 57Fe M\"ossbauer studies were
performed on superconducting Rb0.8Fe1.6Se2.0 with Tc = 32.4 K. The
superconducting transition temperature gradually decreases on increasing
pressure up to 5.0 GPa followed by a marked step-like suppression of
superconductivity near 6 GPa. No structural phase transition in the Fe
vacancy-ordered superstructure is observed in synchrotron XRD studies up to
15.6 GPa, while the M\"ossbauer spectra above 5 GPa reveal the appearance of a
new paramagnetic phase and significant changes in the magnetic and electronic
properties of the dominant antiferromagnetic phase, coinciding with the
disappearance of superconductivity. These findings underline the strong
correlation between antiferromagnetic order and superconductivity in
phase-separated AxFe2-x/2Se2 (A = K, Rb, Cs) superconductors.
