  We study the structural evolution of Sr$_3$Ir$_2$O$_7$ 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
Sr$_3$Ir$_2$O$_7$ at high pressure. In the low-pressure tetragonal phase, we
find that the in-plane rotation of the IrO$_6$ octahedra increases with
pressure. The calculations further indicate that a bandwidth-driven
insulator-metal transition occurs at $\sim$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 Sr$_3$Ir$_2$O$_7$.
