We study the structural evolution of Sr3Ir2O7 as a function of pressure using x-ray diffraction. At a pressure of 54 GPa at room temperature, we observe a first-order structural phase transition, associated with a change from tetragonal to monoclinic symmetry and accompanied by a 4% volume collapse. Rietveld refinement of the high-pressure phase reveals a novel modification of the Ruddlesden-Popper structure, which adopts an altered stacking sequence of the perovskite bilayers. As the positions of the oxygen atoms could not be reliably refined from the data, we use density functional theory (local-density approximation+U+spin orbit) to optimize the crystal structure and to elucidate the electronic and magnetic properties of Sr3Ir2O7 at high pressure. In the low-pressure tetragonal phase, we find that the in-plane rotation of the IrO6 octahedra increases with pressure. The calculations further indicate that a bandwidth-driven insulator-metal transition occurs at similar to 20 GPa, along with a quenching of the magnetic moment. In the high-pressure monoclinic phase, structural optimization resulted in complex tilting and rotation of the oxygen octahedra and strongly overlapping t(2g) and e(g) bands. The t(2g) bandwidth renders both the spin-orbit coupling and electronic correlations ineffectual in opening an electronic gap, resulting in a robust metallic state for the high-pressure phase of Sr3Ir2O7.