A high-pressure (up to 4.95 GPa) single-crystal XRD study is carried out of the natural zeolite-like heteroframework zirconosilicate elpidite |Na1,65Ca0,15(H2O)3| [ZrSi6O15], sp. gr. Pbcm, a = 7.13201(19) A, b = 14.6787(4) A, c = 14.6297(3) A, V = 1531.56(7) A3, Z = 4. At pressures above 1.2 GPa, the elpidite undergoes a phase transition with the doubling of the a parameter of the unit cell (sp. gr. Pbca). The transition is caused by the opposite rotation of the originally equivalent Zr octahedra. Above 3.5 GPa, another phase transition is detected, this time with the symmetry being reduced to monoclinic (sp. gr. C1121/a). The framework deformation leads to a partial reorientation of H bonds in the extra-framework subsystem and a change in the configuration of Na and H2O chain from a unilaterally branched pattern to a bilaterally branched one.