Elastic and structural behavior of a natural cubic analcime (space group: Ia3̅d) from Sardinia (Italy) was investigated at high pressure by in situ single-crystal X-ray diffraction. A first-order phase transition is observed in the pressure range between 0.91(5) and 1.08(5) GPa. Unit-cell constants and reflection conditions confirm that the space group of the HP-polymorph is P1̅. No further phase-transition has been observed at least up to 7.14 GPa. Fitting the volume data of the cubic polymorph with a second-order BM-EoS we obtain: V0 = 2571.2(4) Å3, KT0 = 56(3) GPa, and K′ = 4 (fixed). For the triclinic polymorph, a third-order EoS gives: V0 = 2607(9) Å3, KT0 = 19(2) GPa, and K′ = 6.8(7). Axial bulk moduli of the triclinic polymorph, calculated with a linearized BM-EoS, are: KT0(a) = 29(2) GPa, with K′(a) = 4.9(6) and a0 = 13.727(10) Å; KT0(b) = 20(1) GPa, with K′(b) = 5.2(5), and b0 = 13.751(15) Å; KT0(c) = 11(1) GPa, with K′(c) = 12.6(6) and c0 = 13.822(31) Å. The elastic behavior of the HP-polymorph appears to be strongly anisotropic, being KT0(a):KT0(b):KT0(c) = 2.64:1.82:1.00. The relevant structural variations in response to the cubic → triclinic phase transition are due to tetrahedral tilting. The tetrahedral framework distortion gives rise to a change of the eight- and six-ring channels ellipticity and of the extra-framework topological configuration: it appears in fact that for the high-pressure triclinic polymorph the coordination number of some of the Na atoms is seven (2H2O +five framework O atoms) instead of six (2H2O + four framework O atoms).